Is it correct to say?

Yes. Grammatically correct, if that is what you're asking? And it may depend on the context, too. It seems a little awkward but acceptable.

So depending on the context also what might be alternatively correct is "I thought that she might want me to stay."

Yes though usu we'd skip "that"

'We' might not. It is one of a few responses that are acceptable. Also, to include the word 'that' does not add any more noise to the sentence.

Indeed 'We' mightn't

That it is

In a certain sense tho it does Rag inasmuch as the "thought that she" is sort of tongue twister, like "certain sense tho" or "inasmuch as the"

Probably depends on the context:

"Are you going back to New York or staying at your daughter's for a few days?"
'I'll leave Tuesday. I think that she might be wanting me to stay"

"Mary really wants you to stay more days to help out with the family."
"Yes, I thought that she might be wanting me to stay"