As mentioned previously, there is really not much dining-wise near the airport hotels. If you are coming in to town (not much of a drive from airport to downtown) for Thanksgiving dinner, you may as well just stay in a more central location. As such, I would re-think spending one night in an airport hotel and would suggest you just stay one place the whole time you are here. (Unless you want to spend a night at the Hyatt Lost Pines, midway between the airport and Bastrop and eat dinner there.) As for dinner,
Threadgill's might work well with your schedule.
I would strongly recommend renting a car. Austin is not really a mass-transit oriented city.