In the summer you need to go to the mountains. I recommend some of the campgrounds up in the Boulder Mountains down in southern Utah, or the La Salles above Moab, or the Eastern Uintahs. All are great areas with less pressure than the normal spots.
In the fall I highly, highly recommend the campground next to the San Juan river just outside of Bluff. It's a great home base to go see Natural Bridges NM, Hovenweep NM, Monument Valley, Four Corners, etc.