Depending on what is going on in town that week you may already find it difficult to make reservations for multiple vehicles at the same park. Moab has been getting really crowded lately. I'd go to the we at sites like DiscoverMoab to look up campgrounds and start calling. They have lists of pretty much all the campgrounds and show whether they have cabins or not. Keep in mind that it tends to start getting rather warm in Moab in June so you'll want elec for sure if possible. It does tend to cool off a great deal at night but I think you'll find using Air Conditioning desirable.
Being as how you're willing to drag you jeep down there I'm assuming you rather enjoy the hard core 4 wheeling thing. You can easily spend a month or more exploring down there and not get bored. I've spent months exploring and destroying trucks down there over the years and still really look forward to when I can make it. I now use a side by side atv though as it's less prone to needing expensive repairs after my adventures. You'll find a lot of info on the web listing trails, and rating them as to difficulty. My recommendation is start out on the easier ones for the first day or so and work up till you reach the limit of your rig and driving skill.