We did a similar trip, but with intermittent camping. We started in California, heading northeast through Yellowstone, Mount Rushmore, down through Colorado, Utah, and home.
Although it is driving season, there are not as many people on the road as you might think due to fuel prices. The only time we made reservations was in Yellowstone. We would check in at the campground we wanted to use to reserve our spot, then come back to it at the end of the day when the campground was full. I would imagine the same technique would work at the hotels in Yellowstone.
I have heard motels along the Blue Ridge Parkway can get booked up. You may want to call ahead to reserve a room for that night, and make that your destination for the day. Booking farther ahead than that locks you into a schedule, and doesn't allow you to check out things you hear about from the locals.