We typically camp in this area 2-3 weeks a year (it's just over 3 hours away), so let me give you some insight on the public CG situation.
No fires allowed at Dead Horse Point (charcoal only).
The NFS CG's up in the LaSal's are gorgeous but for small rigs only.
If you arrive decently early on a Monday, you should be able to find a BLM spot, although with your rig (36' isn't it?), you will be limited to the BLM campsites you will fit in. If you arrive around 10-ish, you might luck out and find an open spot at the very popular Goose Island which has some larger sites on the river. Above there is Granstaff CG (where we often end up) but you might walk it first to make sure you can get through the loop at the end. (There's a shoulder to pull off on the river side just beyond where the entrance to the "Porcupine Rim Trailhead Parking") Further upstream, Hal Canyon might have a couple of spots you could fit in and the CG is a loop road. (Some of the other CG's along there are pretty tight to turn around in at the end of the spurs). Just beyond Hal Canyon, Big Bend is one of the larger CG's out there and has some decent sized sites. (But watch out for Upper Big Bend CG as it is another one with very close quarters to turn around at the end). If you're interested in these CG's, check out satellite imagery (Google, etc.) to see what these interiors roads look like. If you find nothing by Big Bend, I'd turn around and head back down stream as the remaining CG's are quite a ways further.
If these are full, then head up to Horsethief which tends to fill later in the day. It's a bit further from many of the attractions and isn't by the river like the more popular ones. But higher up it is a little cooler. I prefer Horsethief to Ken's Lake during the summer, as most of the campsites at Ken's Lake have little shade and little protect if the wind kicks up.
The BLM has just put in over 100 campsites at the Ledges downriver from Moab along the Kane Creek road. Some of these are quite large sites, but the downside is they are 8-10 miles from Moab on the Kane Creek Road and the last few miles are graded but often washboard-y. But they are the last sites to fill. We haven't been out there in the last year, so maybe the road is being better maintained now that they are done with the CG construction.
Except for maybe a couple of campsites, there are no big-rig friendly sites at any of the other CG's along Kane Creek Road nor along Potash Road.
List of BLM CG's in the areaWe've never camped there, but there are 120 campsites at the
Sand Flats Recreation Area. This area is used by mountain bikers and OHV enthusiasts so sites might get full quickly, I just don't know. The area is pretty close to town.