List of AZ shooting ranges, AZ Public Shooting Ranges or AZ Fish & Game list of Shooting Ranges.
Plus, shooting on BLM/public lands might be an option. Check AZ BLM web site: click. Find a spot with a good backstop hill, fire away, clean up before leaving. Makes for a very nice afternoon, without all the range-type restrictions, risky shooters, etc. If uncertain where local spots might be, I'm sure some of the old timers at your range would know of a couple decent nearby locations suitable for shooting on public lands.
It sounds like shooting a shotgun is much like shooting a .50 cal BMG. Snug it up tight, get a firm seat, then just go with the push. (of course, I'm standing with the shotgun, but the same idea applies).