Horses, Welcome to the ACA Forums!
Unfortunately, most of my favorite places in AZ & NM are at higher altitude so while it will probably be warmer than your hometown in January, it will still be cold and snowy at times so most are good in the winter.
In AZ, I like Canyon de Chelly National Monument (may be closed due to Covid); Sedona; the ride (on gravel) between Chiricahua Natl Mon. & Portal is very nice but I "think" there was a forest fire in the area a few years ago; and the ride (partially on gravel) between Patagonia and Sierra Vista makes you think you are not in Arizona at times (see pics).
In NM, I like the Geronimo Trail (partial gravel) between Douglas, AZ, and Animas, NM; the White Sands Natl Mon.; The immediate area around Cloudcroft (though the surrounding areas is not much to write home about); areas west of Los Alamos; and most anywhere in northern central NM.
You might consider riding Baja California, especially on the eastern side of Baja. I have heard that MEX-5 is now fully paved and it is a lot quieter than the western MEX-1. You could also do a ride around the Copper Canyon west of Chihuahua. If you want flat, then the Yucatan & Belize would work.
Sorry could not help you with better winter riding areas.
Tailwinds, John