Cathy, we are up near Joliet. So it takes us the better part of a day just to get through the state. We got advised by friends who had been going to AZ for the winter in their RV. Their advice was not to plan on going I-40. It might be possible, but you have to watch the forecasts and be ready to drop straight south whenever the next day was going to have snow or ice. In fact we have only used I-40 once, when it was so cold everywhere that we had to stay in motels until we reached Las Cruces NM. The southerly route through TX takes about another day. If possible, on the third stop, we take a day off.
As far as AZ, so far we have stuck to the Phoenix and Tucson areas. We like state parks and the Maricopa county/regional parks, which are just like state parks if not better. The latter book up in advance, so you might not have any luck there on short notice except for their overflow sites. Some of those are nice, some are not. The overflow at Cave Creek is good enough to be worth stopping in a pinch. The park itself is great, which is true of all of them. Sites are huge. You can cancel reservations if need be. You might have some luck getting into Tank Mountain Park on short notice. Prior to last year, they did not have a dump station, which held down the demand. It might take a few years for word to get around.
Here is a website for the regional parks:
We have stayed at Catalina SP by Tucson. Half the sites have electricity and water, half do not. If you are willing to dry camp, you can get in on short notice. Same thing at Lost Dutchman SP by Phoenix.
If you want to be in a commercial RV campground, there are many. One that has had openings when we needed them was The Arizonian RV Resort, which is way out at the end of Gold Canyon, south of Apache Junction. It's beautiful desert out there. You can go walk in the desert right from the park. We would walk our dog in the desert.
There are parks in the Casa Grande area, right between Tucson and Phoenix. We have not tried that area yet. Palm Creek RV Resort is highly regarded. Bottom line, without reservations, you might have to hunt a bit, but spots are out there.
Once you try a little dry camping, you might find it very agreeable, because it opens up a whole world of opportunities out there. There is a huge network of pretty CGs at Roosevelt Lake, all are dry camping. As in the state parks, there are good restroom and shower buildings.
Then there is Lake Havasu, Quartzsite, and Yuma, to name the other big areas. We have not even made it out to them yet. My impression is that you could always find a commercial spot in Yuma. Someone correct me if I'm wrong.
Really, the attractions of AZ versus FL are the dry sunny days, the desert experience, and feeling of wide open spaces. South FL is warm when no place else is, is more crowded which also means a more familiar type of civilization. AZ is more about getting away to someplace warm, sunny, outdoorsy, and completely different.