We did an 18 day trip last summer from Ohio - so you're a little farther. We did 3 1/2 long days driving out, 2 1/2 days at Grand Teton, 5 days at Yellowstone, spent a day driving from Yellowstone to Devil's Tower with a stop at Little Big Horn on the way; three days at Custer/Mount Rushmore over the 4th of July and then worked our way home, stopping at Wall, Badlands, Corn Pallace, Laura Ingalls Wilder Home and Clear Lake Iowa on the way home tobreak up the drive a little. Four of those days were really long days, but worth it to have the extra day or so at other locations. I'd do it again - of course I'd love to have a day or two longer, but there's only so many vacation days in a year!
I agree that throwing in Glacier is too much.
We went end of June, beginning of July last summer. Book your National Park camp sites for those big parks now, especially if your rig is big. We stayed at Colter Bay and then Fishing Bridge to have hookups because we're pretty small. There are a lot of other beautiful campgrounds with less (or no) hookups, but you need some capacity for them. (At Rushmore we stayed at Rafter Bar J if you ever do that trip, good spot).
Enjoy - it's a great trip however much you decide to tackle!