I live in Oregon and travel/camp all over the west. I don't believe I've ever made a campground reservation. Aside from showing up at canyonlands national park and secure three sites on the white rim trail anyway. Then again I try to stay away from the huge touristy campgrounds with swimming pools and such.
I don't know where you're going on the north rim but toroweap campground is amazing if your rig can stand a bunch of washboard roads.
The southwest is going to be extremely hot in June too.