We just got back from a month long road trip with the grandkids. NH-CO-UT-AZ-TN-NH, with stops along the way. I made advance reservations for campgrounds in areas we were staying for awhile. But didn't make, nor need reservations for any of the stops out to CO or to any back from AZ (except Gatlinburg). We have never had a problem finding a place to overnight while traveling. We have made reservations in advance for those popular tourist areas we wanted to see.
Having said that we could have gotten into almost all the campgrounds we had made reservations at.The only one that was full was Wahweap at Glen Canyon National Recreation area. Glad we had a reserved spot there.