The BLM will be in more open country and there are also wildlife management areas where you can pull off the road and camp anywhere within a given distance from the road and are self-contained. Go to the BLM website to find locations and information on facilities at them.
Between Phoenix and Flagstaff there are three national forests and these will have campgrounds. Some will be "primative" and only have pit toilets and others may have showers and water for your MH.
What works remarkably well and is easy to do is lookup a city like Flagstaff that is on your planned route home and do a search nearby for "camping" which will bring up campgrounds as well as the commercial RV parks.
Often I have found excellent free publications for camping at the local visitors bureau in a town and often in hotel lobbies. I don't have to be a guest to go inside and grab a publication from the special rack for brochures on local attractions.