We use two basic winter routes from the south suburbs of Chicago to Phoenix. Neither one is the direct route, because of the likelihood of snow and cold.
First choice, and only if the weather is unseasonably good: St Louis-Tulsa-OKC-Amarillo-Las Cruces- Tucson-Phoenix. Once we took this route because the weather was too cold to camp anywhere, even way south, but the sun was shining. We stayed in hotels through Amarillo.
The Amarillo-Las Cruces segment can be one of two ways. Via Portales-Roswell-Alamogordo, or Tucumcari-Santa Rosa-Alamogordo. I prefer the Roswell route, although it's not much difference other than scenery. Ruidoso is a known speed trap.
Typical route: We are forced down south. I-57 to Sikeston-Texarkana-Dallas-& on west to Las Cruces etc. Usually we can avoid snow, and are able to camp, on this route. This is I-57 to 55 to 40 to 30 to 20 to 10.
🙂Once we had to cut far south of Dallas, across Waco & Junction TX. There is a nice little dark sky state park just outside of Junction, and from there it's a good day's drive to Las Cruces.
If you take I-44 & 40 across MO and OK, have a backup plan and watch the weather ahead of you. Be prepared to drop south and get out of the weather before turning west again. Storms can come up fast and furious in the panhandle and mountains, and then you are stuck in the middle of nowhere.
Currently RV-less but not done yet.