The authors of English Language Today, available on the Cambridge Dictionary website, agree with Hewings that in reported speech modals such as could, would, should, etc. admit no change. They do, however, suggest an exception, "especially where the report looks back to a hypothetical event in the past":
"The noise might be the cats next door."
He said the noise might have been the cats next door.
The may also occur when could is used in the same sense as might:
“That could be Mr. Brown's noisy Ford.”
He said it could have been Mr. Brown's noisy Ford.
They offer this example:
He said he would have helped us if we’d needed a volunteer. (original statement: ‘I’ll help you if you need a volunteer’ or ‘I’d help you if you needed a volunteer.’)
The sentence in indirect speech attempts to do double duty for one conditional that leaves open whether the man rolled up his sleeves and a second where he most likely headed off to the pub. In the first sentence, help is contingent on the need for a volunteer; in the second, there is no such need. This is an important distinction that should be transmitted correctly in reported speech, and indeed easily done, since the first conditional should be rendered:
"I'll help you if you need a volunteer."
He said he would help us if we needed a volunteer [so we handed him a shovel].
Will goes to would and need to needed according to the standard rule. In the second sentence, where the man himself has determined his help is not needed, reported speech is cast as a contrafactual conditional in the past:
He said he would have helped us if we had needed a volunteer [but we didn't, so off for a quick pint].
And this brings up a problem with your suggested examples: an attempt to change tenses with a modal has resulted without exception in an irrealis:
"I could meet you at the airport [if I get off work in time]"
He said that he could have met us at the airport [but he didn't because he had a late meeting].
You also offer the example:
"You should contact me."
And your suggested transformation:
She said I should have contacted her.
But that isn't what she said.
"You should contact me."
She said I should contact her [so I did at the time she suggested].
"You should have contacted me."
She said I should have contacted her [but I thought she meant next Wednesday].
This leaves might as the sole modal which can permit a change in indirect speech because its meaning of maybe yes, maybe no is preserved. Attempting to do so with other modals completely changes the sense of the original speech unless it was couched as an unreal condition in the first place.