You'd have a much better chance if you could do with out full hookups for at least some of the trip.There are a number of Forest service Campgrounds in Arizona.Some have no hookups,some electric.Also Arizona State parks have water and electric.When you're traveling during prime camping season,campgrounds will be pretty well booked,especially on the weekends.However,if you're flexible with you hookups,then I think it is do able.Check out Suzi and Fred Dows Forest Service Camping website. They have all FS campgrounds listed with all the amenities.
http://www.forestcamping.com/
