X2 on straight south then I-10 into Arizona. I-40 can be very bad in winter.
You could check out Gulf State Park, Alabama - lots of snowbirders go there. You can stay the whole winter, unlike many state parks...nice beach, good food, fresh shrimp off the boats, good shopping
Then work you want into Texas and check out some areas. New Mexico is generally too cold for wintering but a look-see in Las Cruces would be worth a stop...perhaps a day trip to White Sands Nat'l Monument.
First major stop in Arizona should be Benson. Kartchner Caverns has a nice state park and the cavern tour is great. Tombstone and Bisbee are worth visiting.
Then Tucson has some great things to do - Desert Museum, Saguaro Nat'l Park (no camping)
The greater Phoenix area has some great Maricopa County parks. The sites are large with lots of vegetation and nice hiking. Lost Dutchman State Park in Apache Junction is awesome and lots to do in the area, including a drive on the Apache Trail for beautiful scenery with lakes.
Full-timed for 16 Years . . . Back in S&B Again
Traveled 8 yr in a 2004 Newmar Dutch Star 40' Diesel
& 8 yr in a 33' Travel Supreme 5th wheel