I am looking for a verb that describes speaking with a full mouth, like a child asking something at the dinner table before swallowing, because they are so anxious/curious for the answer. Anything come to mind? I didn't find ANY.
Speaking with your mouth full, as a phrase, means speaking with your mouth full of food, but I take it that is not your meaning.
To say something suddenly and without thinking is to blurt.
To speak excitedly and at excessive length is to gabble.
To speak without making sense (due to anxiety, ignorance, or some other reasons) is to babble.
I don't believe that there is a specific word for "talking with food in your mouth", but you can certainly describe how it sounds when you talk with food in your mouth:
"Mom, why do cats chase mice?" Jimmy slurred around a mouthful of peas.
"Mom, why do I have to go to school?" Jimmy garbled, his mouth full of cereal.
Or you can describe the results of the talking-with-stuff:
"Mom, will I ever have a sister?" Jimmy asked, dribbling half-chewed lima beans onto his lap.
"Mom, where do babies come from?" Jimmy demanded as he chewed, flecks of pie spraying out onto the table.
The closest verb I can come up with is sputter
To spit out or spray particles of saliva or food from the mouth in noisy bursts.
The American Heritage Dictionary
From Google Books an example
“What? You made me?” The boy sputtered and almost choked on his corn. He wiped his mouth on his sleeve.