Once the kids are back in school there usually isn't too much problem with finding sites but a couple of your destinations are still pretty popular even year round. If you are looking to be in a specific, popular, campground (like Camper Village at the Grand Canyon as an example) then you may want the reservation. If you just want to see the sites and don't mind camping a bit away from the parks then there should be several openings at that time.
I usually make a reservation at the real popular national parks if I want to camp in the park, but also leave room in between my stops for ease of driving and in case I want to check out a place I hadn't originally thought of, but fulltiming gives me that freedom. Your trip will probably have to be a little more structured if you want to stay right in the parks.