How do I meet the 7 feet high ceiling requirement after finishing the basement?
Per the wording of that code, you can't be adding anything to the floor or the ceiling as you'll be under the 7' minimum. You could consider a super-thin flooring (stain the concrete? Linoleum?) and then, as you suggest, put the sheetrock between the joists (though that sounds like a finishing nightmare).

Alternatively, raise the foundation (likely cost prohibitive).

All that said, I echo B Mitch...what is the concern of not meeting the code? The main drawback would be that you couldn't advertise it as finished square footage when you sell.

The first thing I'd do, though, is have an inspector come out and ask what the odds would be in getting a 1" variance or so. They may count it as 'close enough' and let you get by.