Tentman - I found this campground via the park Web site, and I tent camped there in May 2002.
This is a great campground because you can camp under shady cottonwoods - welcome in this desert setting - near the banks of the Green River with fine views of rocky cliffs that hold dinosaur bones and petroglyphs for exploring. There is lots of bird life along the river. I saw my first Say's Phoebe. The campground has 88 sites for tents or RV's, running water, and flush toilets, but no showers.
They have firewood for sale. Campsites cost $12 a night. In May 2002, the campground had maybe 10 people in it; it was very private.
Kids will enjoy the dinosaur museum in nearby Vernal. The much larger and more sophisticated Prehistoric Museum in Price, UT is farther away, but on your way to the Moab area. This museum has great fossils and early Native American stuff, and it was near-deserted when we visited in May.