Last year, we did a 10K mile trip from New England to Washington and back with a 4 day stint at Yellowstone's Fishing Bridge in June. We made reservation for Yellowstone and found out that we really didn't need to as there were numerous empty sites. Snow was still piled quit high between the sites.
Here is what I would suggest to you, which worked extremely well for us:
We would stay at Cracker Barrels over night and every two nights dump and take on water at Flying J's. Join (for free) their RV club, which gives you 1/2 off dumping fee = $5.00. On our return trip through Denver, did the same thing and never a problem. There are boon-docking opportunities all over this country.
Good luck and don't sweat it, you have a great trip!
