We live in Phoenix and travel up to Flag all summer long, much cooler up there.
Black Barts.....not a very nice camp ground, would never stay there again, however the restaurant is very very good!
Geers....nice little campground very friendly staff and not to far out of town on the North East side of Flag
KOA....have never stayed there but have had some friends that did and would not recommend, depending on the size of your rig can get pretty tight.
Woody Mountain...nice campground if you like rough and rugged. If it rains can get pretty bad in spots.
William's down the road, west on I-40 has some nicer campground, might want to try there.