Trip routing programs will probably suggest that you travel the shortest/northern route (Memphis, OK City, Amarillo, Albuquerque/Gallup, Holbrook/Scottsdale), BUT.....
during winter, I'd highly recommend the lower elevations & Interstate across Texas - you're less likely to encounter snow & ice.
Option A: I-20 to Dallas/Abilene/Fort Stockton; then I-10 to El Paso, TX & Tucson AZ, then Scottsdale. Approx 1850 miles.
Option B: I-85 to Mobile, then I-10 to Scottsdale is approx 1975 miles... a longer trip, but the weather may be even nicer?
Our kids live in TX and AZ, and we travel the 1-20/I-10 route quite often (& year-around). There's good overnight RV parks all along the way, with easy access
to/from the Interstate. I'll gladly give you names of places we've stayed, as I'm sure other members will too.
If you're not familiar with winter RV'ing - keep in mind that it gets dark sooner, and small RV parks in less-populated areas out west can fill up quickly & early!
And remember... a frozen water hose does not roll/store easily
...carry a couple gallons of bottled water, just in case freezing temps are predicted during your overnight stops.
Late March/early April, the desert should be 'in bloom'... should be a pretty return trip!
Hope this helps.
Terri, the Chevy co-pilot & DH Kevin