If I were coming from the St. Louis area in December, I would take I-55 to I-30 at Little Rock to I-20 at the D-FW area to I-10. That would get you south as quickly as possible, but still give you a fairly straight route to southern NM and AZ. Oklahoma is right on the freeze line that time of year and can have sleet/ice storms that you don't want to be driving in. North Texas can also have ice, but generally just has rain that time of year.
There are some very nice Texas state parks with hookups spaced along that entire I-30/I-20/I-10 route. Prices are reasonable and they generally have a lake or river associated with them, so are much more scenic than a KOA. We have used many of them in winter and you won't need reservations. See the Texas State Parks website for more info.
As others have said, it can be in the 40's at night in southern New Mexico in winter, but we still prefer the small-town feel of the Las Cruces area over Phoenix. Phoenix is just too big--we might as well stay in Denver! In Las Cruces, we stay at Sunny Acres RV Park, which is right in town and within walking distance of a grocery store. And it does warm up nicely during the daytime!
Others have mentioned Kartchner Caverns and Benson, AZ, as good spots. We really like the Butterfield RV Park in Benson, which has an observatory with nightly sky viewing programs, a beautiful pool, and a big community center for dinners and other activities. We have met snowbirds there who have come from as far away as Ohio and Saskatchewan every winter for 8-10 years. And, for short-term guests, they have a row of asphalt-paved, full-hookup spaces directly behind the office-pool complex.
If you are at all interested in historic aircraft, DO NOT MISS the Pima Air & Space Museum in Tucson. It probably has the best warbird collection of any museum outside the Smithsonian. It is right off I-10 in southeast Tucson.