Near Moab, try the La Sal Mountains. The west side above Moab has great vistas to the west, while the east side has a huge aspen forest. You can use it as a base to visit all the Arches, Moab and Canyonlands, all of which will be a lot hotter than the mountains as Cayuse noted.
Depending on your route to Boulder, you could try Great Basin NP and Grand Mesa. We pulled into Great Basin (Nevada) four or so years ago mid afternoon in July and not only did we get a campsite, but we had lots of vacant ones to choose from. (On the return leg of the trip, we had to scrub the Sierras due to a fire and thought we would spend the night at a hotel in Death Valley. Seriously, who goes to Death Valley in July? Lots of Germans, no room was to be found.) Great Basin is not well visited. Nice caverns to tour, great park. Grand Mesa, near Grand Junction, Co, has lots of dispersed campsites, though the mosquitos can be nasty in June.
As for SW Colorado, there are tons of dispersed places to camp. Campgrounds might be booked, but dispersed camping is pretty pletniful. Near Silverton, there's Mineral Creek Road, though it gets crowded on weekends and is probably better for tents. There is also Line Creek Road. We have used it as our emergency camp site when we get into the area late. South of Telluride, there's the West Dolores/Dunton Valley and the more popular area around Lizardhead Pass. You can also find nice campsites behind Trout Lake if you take the forest service roads. You will have no problem during the week finding a place to camp, weekends can be far more crowded.
On way from SW Colorado to Moab, you can camp above the Mogi Dugway and visit the Valley of the Gods.
Just remember that some of the places you might be passing through are tribal land and different rules apply there.
