I would not recommend you taking I-40 to Flagstaff the end of December. That highway is known for bad storms especially around Albuquerque. We were caught in one the end of April one year.
Instead, head south to I-20/I-10 and you'll have much better weather. Once you're in Arizona at that time of year you'll need to watch elevations. Basically, you will want to stay in these areas coming from the east: Benson, Tucson, the greater Phoenix area (Apache Junction is nice and 'roomier'), Lake Havasu and down along the Colorado River to Yuma. You don't want to travel in the northern half of Arizona in January.
New Mexico is also very cold - even the southern portion. Depending on your timing to get to Arizona, you might want to check out the Hill Country area of Texas. We like Fredericksburg. It will be cool with some warm days. Big Bend National Park is another place if you have time.
In Arizona, all of the state parks are great and o.k. for big RVs. A few of our favorites are Kartchner Caverns near Benson/I-10, Catalina in Tucson and Lost Dutchman in Apache Junction. Also in the Phoenix area are the awesome Maricopa County Parks with lush Sonoran desert landscaping. Reservations are recommended as they are popular. Gilbert Ray campground in Tucson is also a great place to stay. It's near the Desert Museum which is a must. When you narrow down your route somewhat, come back for specifics on areas.
Have a great trip!