Generally the Sonoran desert areas are a little warmer than the Chihuahuan desert region. You could check the forecast for the lower elevations of the Gila between cold fronts (I know from experience the Middle Fork can go to zero with a strong enough front) or the Superstitions east of Phoenix mostly staying low (the camp for below Reavis Falls was warner than the Reavis Ranch point).
Here's my pic show flowing water but it was cold enough to form ice around the falls.
Above the falls (the former site of the ranch) was colder ..
The lower elevation of the Mazatzal is very desert like northeast of Phoenix/Scottsdale// south of Payson and has access to the Verde River (treat it). Still trying to find good vehicle access from the west but might need to go in from the northeast
In the desert, elevation is key. The higher elevations will start accumulating snow. Also a front will bring cold winds and temps even on the desert floor.
I'm looking at the mountains of Tucson really soon but there could be a little snow at about freezing at the highest elevations.
I prefer sticking to USFS or NPS lands
There's also BLM land in the New Mexico desert but you need to park on federal land, … not the patchwork of private or state (some ranchers will call "the authorities" on a car parked wrong, so position with a GPS and a BLM map from Las Cruces if you are looking at doing this).