Probably THE nicest State Park in AZ is Lost Dutchman, about 30 miles east of Phoenix in Apache Junction. Close to some great riding, many sites but good privacy in the Mesquite & Palo Verde understory, lots of birds, separate tenting sites so the generator noise is not a huge problem. I enjoyed a week there last year. Hot showers too. Perfectly safe.
Another place I enjoyed last year was near Portal in the Chiricahua Mts. Quiet, pleasant CG's and interesting rides just up the road. However, a large area of the Chiricahuas burned last spring, so it might not be so great now. The Nat. Monument fared better and is really interesting, but more crowded and no back roads.
Don't wait too long, most of the lower areas in AZ get pretty hot by June, February to May is best.