I found a great campground very close to Moab. The Oak Grove Campground is located about 7 miles up highway 128 just north of town. It's run by the BLM and costs a mere $12 a night. The camp sites on the left side are sheltered by a group of oak trees providing nice shade from the hot sun. The campground is beside the Colorado River with colorful red rock cliffs towering above.
The BLM operates several campgrounds along highway 128, so I had many choices. I liked this one because of the shade and privacy. I also liked the sites at Goose Island Campground, but it was more crowded.