I have camped and 4wheeled in the San Juans. I saw your post on the mogollon rim but if you are looking to camp near water the San Juans are it. You should map a route so you can take the mountain passes from one town to the other. Go to Wide World of Maps here in Phoenix and get the Uncompadre National forest map. You will see won hat I mean about the mountain passes. Check and see what events are going in Ouray and Telluride. Remember, it will be cold at night and it will rain.
So many great places down there, it's difficult to say for sure. Depending on the snow pack, some of the passes won't open until late July (although they were all open early this year). Anywhere around Ouray or Silverton is amazing, just have to wheel up a little (most trails are pretty easy) and find a good spot. Our Colorado book covers the area in full detail
It depends on which pass. We did Ophir Pass on our way to Telluride and it took maybe 2-3 hours. Or you can take Black Bear Pass, which I haven't done yet. Then you can take Imogene Pass out of Telluride over to Ouray. That one can take a while and you may be dealing with snow.
Then I would base out of the Ouray area and wheel from there. Black Bear pass is a one way, going toward Telluride so you go over from the Ouray side into Telluride and then take Imogene pass from Telluride back to Ouray.