I can't speak as to whether a place has dust or not. We're in a drought and pretty much all of WY and MT is a bit dustier this year.
But, I do recommend staying at a lodge and renting horses as opposed to bringing your own. I live in WY and happen to meet people who have done it both ways.
The downside of bringing yours is: no local vet if something happens, you don't have a farrier, they aren't conditioned to the altitude and terrain, you don't know which way to go in the few hundred thousand acres where you'll be riding, there are numerous regulations on what you can feed, where you can feed and how you can feed.
I know ranchers who take dude ranch vacations. Several couples will meet in a part of the state where they don't live and work just for a get away. They get to enjoy all the fun parts without the work and worry that most don't know is part of ranching. Plus, a cook and a pool!
Don't discount the mountain areas beyond Yellowstone and Glacier. CO, WY, and MT have numerous other beautiful ranges. They will be less crowded and less pricey, while still providing a fantastic experience.
Have fun whatever you do.