Here's a link to AZ State Parks:
Arizona State ParksNo, there is no camping pass for Arizona. New Mexico does have one but New Mexico during the winter months is very cold. We were going to try it one year and only lasted a couple weeks because of the cold and wind.
We love all the parks so it's best to try them all. We especially like Lost Dutchman in Apache Jct. (we volunteered there). Kartchner has the caverns to tour. Dead Horse Ranch in Cottonwood is great and it's easy to tour Sedona from there but Dec-Feb can be cold. Alamo Lake is remote but very pleasant (good paved road). You really can't go wrong.
You also might want to try the Phoenix Regional Parks which are like state parks - very scenic. Over by Roosevelt Lake are some extremely nice forest service campgrounds - Windy Hill and Cholla are great.
Have a great trip!