South and east of Tempe has some great 4WD accessible canyons. One I think is Cottonwood Cyn which had some of the oldest and largest cottonwoods I have ever seen. East and North of Phoenix have some great back roads and wonderful desert beauty. South of Florence Jct. are some great camping and jeeping areas. Just wide open BLM land.
About staying anywhere near the southern border: don't. Much of southern AZ is closed to Americans now with so much human and drug trafficking taking place. We tried to camp near Organ Pipe a few years ago and the gates were down and there was a FED sign saying keep out on account of desperate people migrating. As close as we could get was Ajo.
jefe