sgrizzle,
Not going to find much "boondocking" around Moab. The BLM has closed a lot of dispersed sites but does have some designated campgrounds with a small fee near Moab.
If you stay in a national forest see if you can get a MVUM map (free at the local ranger station). These maps will show you what roads are open and where you can camp and how far off the road you can camp. If you have a smart phone get the Avenza PDF map program. You can download forest service maps (again for free) that show what roads are open and where you can camp. Not sure if all forests are available but I do know many of the forest in Arizona are using it. Also keep in mind that just because a road has a number with a sign does not mean it is open, you have to refer to the map to verify it the road is open or not.
As far as BLM goes, I have not seen anything consistent from them. It varies all over. Typically you can camp just about anywhere unless its marked that you can't. The more active a area is the more restrictions in place.
I will also use google maps with satellite images to find camping sites. However with pine forests it is hard to find those dispersed camping sites as the trees cover the sites.