Make Yellowstone & Tetons reservations NOW. The parks are less crowded than in midsummer, but they are by no means emptied out. In addition, some of the CGs inside the park close after Labor Day. (Outside the park most are likely to remain open at least through most of September). Glacier's season usually ends fairly early, so be sure to check on season closing dates there as well (and I'd recommend reservations there too.) Good CGs for Yellowstone: the only CG inside the park with any hookups is Fishing Bridge, which is centrally located, but sites tend to be tight. (On the other hand, you'll probably spend all day & into the evening driving around the park to see the sights.) Outside the park, we like Grizzly RV Park in West Yellowstone; it's close to the west gate of the park, very nice, clean & plenty of room for all sizes of rigs.
Since we need hookups, we have chosen Colter Bay RV Park for the Tetons. For your size camper, if you can deal w/o hookups, you might consider Colter Bay or Gros Ventre CGs (not to be confused with the RV park) - Signal Mt CG will be too small for your camper, however. (Sites too short.) IMO there aren't any other private CGs that pass muster in GTNP and Jackson - of course, that's our personal taste.
If you can, be sure to allow at least 3 full days for Yellowstone and 1 full day for the Tetons. If you are active & enjoy hiking & other outdoor activity, add a day (minimum) to each park. For Glacier, I'd recommend allowing at least 2 full days, and 4 would be better. If you can, divide your time (and your campground bases) between west & east side of Glacier; they are very different from each other. Remember you cannot pull your trailer over Going to the Sun Road; you'll have to go around, on US2. It's a nice drive too, but doesn't compare. The red bus tours, if still operating when you are there, are wonderful, worth every penny, IMO.
Have a fun trip!