Compound predicates can sometimes take, and indeed really must have for correct understanding, a comma separating them even though no new clause is begun.

Examples from Tolkien:

• Its explanation lies in the history of the Ring, as it was set out in the chronicles of the Red Book of Westmarch, and is now told in The Lord of the Rings.

• He had decided that he was not quite his sort, and wanted him to go away.
• ‘Come along in, and have some tea!’ he managed to say after taking a deep breath.
• Then they went back, and found Thorin with his feet on the fender smoking a pipe.
• The few of us that were well outside sat and wept in hiding, and cursed Smaug; and there we were unexpectedly joined by my father and my grandfather with singed beards.
• People would see if he would stand being kicked, and driven into a hole and then robbed.
• When he was found he had already been there long, and was on his way back.
• Be sure that he took so little hurt from the evil, and escaped in the end, because he began his ownership of the Ring so.
• At first Frodo was a good deal disturbed, and wondered often what Gandalf could have heard; but his uneasiness wore off, and in the fine weather he forgot his troubles for a while.
• Merry took charge of this, and drove off with Fatty (that is Fredegar Bolger).
• After a rest they had a good lunch, and then more rest.
• ‘Yes, it is all very dim, and stuffy, in here,’ said Pippin.
• Holding the hobbits gently but firmly, one in the crook of each arm, Treebeard lifted up first one large foot and then the other, and moved them to the edge of the shelf.
• Merry and Pippin climbed up the path that came in from the west, and looked through the opening in the great hedge.
• He wrapped himself again in his old tattered cloak, and led the way.
• Nonsense maybe, and maybe not.
• Pippin gazed in growing wonder at the great stone city, vaster and more splendid than anything that he had dreamed of; greater and stronger than Isengard, and far more beautiful.
• Bergil clapped his hands, and laughed with relief.
• ‘There and elsewhere in many places,’ said Faramir, and sighed.
• Denethor turned his head slowly from Faramir’s face, and looked at them silently.
• He sprang to his feet, and pressed himself against the wall beside the road.
• Hardening his will Sam thrust forward once again, and halted with a jerk, staggering as if from a blow upon his breast and head.
• And Aragorn gave to Faramir Ithilien to be his princedom, and bade him dwell in the hills of Emyn Arnen within sight of the City.
• ‘Well, no one troubled us,’ said Pippin, ‘and we came along slowly, and kept no watch.’
• But Sam turned to Bywater, and so came back up the Hill, as day was ending once more.

The Wikipedia like you gave is actually unrelated to the matter at hand. It is talking about separating clauses. When you have two independent clauses connected by a coördinating conjunction, it is considered “mandatory” to use a comma before that conjunction.

• I need to run down to the store down at the corner, and you should come with me.

The only exception generally admissible to this rule is when the two clause are especially short:

• Either I leave or you do.

However, in your own example, the subject does not change. That means you have a compound predicate. You do not have two separate clauses.

• I need to run down to the store and grab some milk.

Notice that that and is coördinating two verbs governed by the same subject, and as such, does not have to have a comma. However, you will note that in the previous sentence, the same situation of one subject and two verbs applied, but this time I did use a comma nevertheless.

That’s because compound predicates can sometimes take, and indeed really must have for correct understanding, a comma separating them even though no new clause is begun. A comma there is not forbidden, but it takes a good ear for the language to know when to do this, and when not to. Placing a comma before an and that is not separating clauses is tricky, and somewhat open to taste and judgement, and disagreement.

Here is a fine set of examples of commas before a coördinating conjunction from one particular Professor of English, someone who certainly had an ear for the English language.

• Nonetheless Ulmo loves both Elves and Men, and never abandoned them, not even when they lay under the wrath of the Valar.
• Nevertheless he met Fingolfin before the throne of Manwë, and was reconciled in word.
• Its explanation lies in the history of the Ring, as it was set out in the chronicles of the Red Book of Westmarch, and is now told in The Lord of the Rings.

• He had decided that he was not quite his sort, and wanted him to go away.
• ‘Come along in, and have some tea!’ he managed to say after taking a deep breath.
• Then they went back, and found Thorin with his feet on the fender smoking a pipe.
• The few of us that were well outside sat and wept in hiding, and cursed Smaug; and there we were unexpectedly joined by my father and my grandfather with singed beards.
• People would see if he would stand being kicked, and driven into a hole and then robbed.
• Bilbo’s heart thumped every time one of them bumped into another, or grunted or whispered in the dark.
• That you should neither drink of, nor bathe in; for I have heard that it carries enchantment and a great drowsiness and forgetfulness.
• When he was found he had already been there long, and was on his way back.
• Be sure that he took so little hurt from the evil, and escaped in the end, because he began his ownership of the Ring so.
• If I say he is a Burglar, a Burglar he is, or will be when the time comes.
• At first Frodo was a good deal disturbed, and wondered often what Gandalf could have heard; but his uneasiness wore off, and in the fine weather he forgot his troubles for a while.
• Merry took charge of this, and drove off with Fatty (that is Fredegar Bolger).
• After a rest they had a good lunch, and then more rest.
• And even so he would never have just forsaken it, or cast it aside.
• But when these were foiled in Bree and at Crickhollow, they returned to their Captain with tidings, and so left the Road unguarded for a while, except by their spies.
• ‘Yes, it is all very dim, and stuffy, in here,’ said Pippin.
• But I have a deed to do, or to attempt, before I too am slain.
• Holding the hobbits gently but firmly, one in the crook of each arm, Treebeard lifted up first one large foot and then the other, and moved them to the edge of the shelf.
• You may know, or guess, what kind of a tale it is, happy-ending or sad-ending, but the people in it don’t know.
• Merry and Pippin climbed up the path that came in from the west, and looked through the opening in the great hedge.
• He wrapped himself again in his old tattered cloak, and led the way.
• Nonsense maybe, and maybe not.
• Even if you chose for us an elf-lord, such as Glorfindel, he could not storm the Dark Tower, nor open the road to the Fire by the power that is in him.
• It belongs to Sauron and was made by him alone, and is altogether evil.
• For he is very wise, and weighs all things to a nicety in the scales of his malice.
• Finish your book, and leave the ending unaltered!
• Pippin gazed in growing wonder at the great stone city, vaster and more splendid than anything that he had dreamed of; greater and stronger than Isengard, and far more beautiful.
• Bergil clapped his hands, and laughed with relief.
• ‘There and elsewhere in many places,’ said Faramir, and sighed.
• Denethor turned his head slowly from Faramir’s face, and looked at them silently.
• Why wait till now, and go through all the labour of the climb, and come so near the land he fears?
• Then Frodo stepped up to the great grey net, and hewed it with a wide sweeping stroke, drawing the bitter edge swiftly across a ladder of close-strung cords, and at once springing away.
• He sprang to his feet, and pressed himself against the wall beside the road.
• Hardening his will Sam thrust forward once again, and halted with a jerk, staggering as if from a blow upon his breast and head.
• I saw him no more, and know no more.
• And Aragorn gave to Faramir Ithilien to be his princedom, and bade him dwell in the hills of Emyn Arnen within sight of the City.
• ‘Well, no one troubled us,’ said Pippin, ‘and we came along slowly, and kept no watch.’
• But Sam turned to Bywater, and so came back up the Hill, as day was ending once more.

Tolkien was extremely careful with his punctuation, and knew what he was about. Yes, some of those commas you might get away without, but surely no one can call them “wrong”. In the same way, whoever or whatever is calling your original example “wrong” is mistaken. Your comma there is just fine, and will do.

Compound predicates can sometimes take, and indeed really must have for correct understanding, a comma separating them even though no new clause is begun.

Examples from Tolkien:

• Its explanation lies in the history of the Ring, as it was set out in the chronicles of the Red Book of Westmarch, and is now told in The Lord of the Rings. * He had decided that he was not quite his sort, and wanted him to go away.

• ‘Come along in, and have some tea!’ he managed to say after taking a deep breath.
• Then they went back, and found Thorin with his feet on the fender smoking a pipe.
• The few of us that were well outside sat and wept in hiding, and cursed Smaug; and there we were unexpectedly joined by my father and my grandfather with singed beards.
• People would see if he would stand being kicked, and driven into a hole and then robbed.
• When he was found he had already been there long, and was on his way back.
• Be sure that he took so little hurt from the evil, and escaped in the end, because he began his ownership of the Ring so.
• At first Frodo was a good deal disturbed, and wondered often what Gandalf could have heard; but his uneasiness wore off, and in the fine weather he forgot his troubles for a while.
• Merry took charge of this, and drove off with Fatty (that is Fredegar Bolger).
• After a rest they had a good lunch, and then more rest.
• ‘Yes, it is all very dim, and stuffy, in here,’ said Pippin.
• Holding the hobbits gently but firmly, one in the crook of each arm, Treebeard lifted up first one large foot and then the other, and moved them to the edge of the shelf.
• Merry and Pippin climbed up the path that came in from the west, and looked through the opening in the great h edge.

Compound predicates can sometimes take, and indeed really must have for correct understanding, a comma separating them even though no new clause is begun.

Examples from Tolkien:

• Its explanation lies in the history of the Ring, as it was set out in the chronicles of the Red Book of Westmarch, and is now told in The Lord of the Rings.

• He had decided that he was not quite his sort, and wanted him to go away.
• ‘Come along in, and have some tea!’ he managed to say after taking a deep breath.
• Then they went back, and found Thorin with his feet on the fender smoking a pipe.
• The few of us that were well outside sat and wept in hiding, and cursed Smaug; and there we were unexpectedly joined by my father and my grandfather with singed beards.
• People would see if he would stand being kicked, and driven into a hole and then robbed.
• When he was found he had already been there long, and was on his way back.
• Be sure that he took so little hurt from the evil, and escaped in the end, because he began his ownership of the Ring so.
• At first Frodo was a good deal disturbed, and wondered often what Gandalf could have heard; but his uneasiness wore off, and in the fine weather he forgot his troubles for a while.
• Merry took charge of this, and drove off with Fatty (that is Fredegar Bolger).
• After a rest they had a good lunch, and then more rest.
• ‘Yes, it is all very dim, and stuffy, in here,’ said Pippin.
• Holding the hobbits gently but firmly, one in the crook of each arm, Treebeard lifted up first one large foot and then the other, and moved them to the edge of the shelf.
• Merry and Pippin climbed up the path that came in from the west, and looked through the opening in the great hedge.
• He wrapped himself again in his old tattered cloak, and led the way.
• Nonsense maybe, and maybe not.
• Pippin gazed in growing wonder at the great stone city, vaster and more splendid than anything that he had dreamed of; greater and stronger than Isengard, and far more beautiful.
• Bergil clapped his hands, and laughed with relief.
• ‘There and elsewhere in many places,’ said Faramir, and sighed.
• Denethor turned his head slowly from Faramir’s face, and looked at them silently.
• He sprang to his feet, and pressed himself against the wall beside the road.
• Hardening his will Sam thrust forward once again, and halted with a jerk, staggering as if from a blow upon his breast and head.
• And Aragorn gave to Faramir Ithilien to be his princedom, and bade him dwell in the hills of Emyn Arnen within sight of the City.
• ‘Well, no one troubled us,’ said Pippin, ‘and we came along slowly, and kept no watch.’
• But Sam turned to Bywater, and so came back up the Hill, as day was ending once more.