My sister lives in Cave Creek and I've gone out there for Thanksgiving a number of times.
My wife, daughter, and I rent through VRBO and have stayed in a lot of locations between Scottsdale and Carefree.
My best advice for a place to stay is to look at the listings on their website and see what appeals to you.
A restaurant in Care Free that we try to go to during our stays is El Encanto, Mexican food with a great pond.
It is a tradition and we have been going for over 20 years.
There are short walks available all through the area, numerous sight seeing opportunities, and touristy type shops you won't see at other towns.
Relax, and enjoy. Happy anniversary!