I know the Swell well, having lived in Green River for some time. The roads are not really all that RV friendly, though are fine for smaller trailers and pickup campers. The problem is ruts and washboards. It's also become an ATV paradise - and I do mean they are everywhere, which is fine if you like noise and dust (I don't).
Having said that, depending on what you're driving, you can go in from the Huntington side and camp at the Wedge, which has a primitive campground with stunning views (watch your kids and pets).
Buckhorn Draw has good camping by the San Rafael River, also a few sites on up the canyon, though these are usually taken.
If you go in by the Temple Mtn side (near Hanksville), you can drive on up onto the Swell and camp anywhere you want. Ditto for the road coming off the freeway (I-70), which meets up with the Buckhorn Draw rd, lots of camping there. It meets up with the Temple Mtn road going the other direction.
The Swell is not nearly as primitive as it was just a few years ago, with the ATV crowd everywhere. There are some great day-trips into places like Hondu Arch and Red Canyon, but most aren't good roads for RVs.
It's a big area and you can even come in on the Green River cutoff and find camp spots, but I wouldn't recommend it for anything but smaller rigs.
And be warned - as you know, most of the Swell becomes impassible when wet. Watch the weather.
Having said that, if you have a smaller rig, the Swell is full of great camp spots. Just pick a road and head out, you'll probably find something. Lots of wild burros.