We have done that kind of trip often leaving our TT at home for the 4-6 weeks.
Check all of the NFs and BLMs in the areas you want to go to. Send for infomation from each state and see what your library has. Buy the books that look like they have good information.
And ask each state for information on their state parks.
We started out covering lots of states in two weeks to spending more time at each stop, seeing more and enjoying ourselves more.
Near Las Vegas, NV - Toiyabe NF's Spring Mountain National Recreation Area, any one of their campgrounds but Dolomite is a personal favorite
Between Yellowstone and Teton NP - Targhee NF's Cave Falls cg - best drinking water around and at an undiscovered entry to the Park
Grand Canyon's North Rim - Kaibab's De Motte Park cg - the wildlife viewing is super and convenient to the Park however we spent most of our time exploring the area and visiting places like Dog Point and Marble Canyon viewpoint
Yosemite NP - Tioga Lake cg in the Inyo NF - Some of the best views of the park are from this little campground
Near Bryce NP - Red Canyon cg in the Dixie NF (really high marks for hiking and history - this was Butch Cassidy's old neighborhood)
Note all are in national forests so the crowds won't be as huge a problem.