We did a similar route in August and September, though we chose Western Colorado. Never any problem finding diesel using Gas Buddy for best prices. (We do not tow, but others have covered good advice on that aspect.)
We usually go for state and National Parks. We overnighted at Badlands NP, CG on site now has electric...then based ourselves at Custer State Park's Game Lodge CG for five days and enjoyed it a lot. Bison herds came past the fence every day, and there are lots of things to do in the area, including Rushmore, Wind Cave, Jewel Cave etc.
We spent a night at the Devils Tower's CG, dry camping, loved it; two nights near the Little Big Horn battlefield; and three nights in Cody to enjoy the town and museums. From there, we entered Yellowstone via east entrance and stayed five nights in Fishing Bridge (book early) with hookups. Don't miss Grand Teton NP. We stayed in the park at Colter bay RV for three nights, pricey but hookups and services. Snake River raft trip is great fun.
Feel free to send me a private message if I can answer any questions about campgrounds etc.