Lake Pleasant - Maricopa County Park has primitive camping for $15.
https://www.maricopacountyparks.net/...-camping-area/
Other Maricopa County Parks: Cave Cree & Usery
I think I read that last week some national forest campgrounds were closed north of Usery because of flooding but that may have been lifted.
There a lot of national forest up in Payson if you're heading that way and also some from Apache Jct. eastward.