I've camped way up Kane Creek Road in some undesignated camping, and the BLM maintains some campgrounds prior to that. The page below shows their campgrounds in that area, and the one I'm talking about is south of the Hunters canyon one, up in a wide open area. Stuff may be fairly snowy right now though, which may mean you want to try to poach the ones by the river. Cane Creek Road, if you're not familiar, is accessed on the south side on town by turning west at the McDonalds. The below pic makes it look like moab is south, but the downtown is north of the turn. They highly regulate/patrol this area in the summer. Winter...?
I've also seen some undesignated camping (but never tried to camp there) out on the Needles overlook road north of the Indian Creek turn off. Its west of the highway, passed Wind Whistle Campground, which is maintained only til October. Its road 133 on the map below.