In my limited experience (about 100 nights on the road since buying my current coach in 2016), the only time you may have issues are the major holiday weekends, or when near major tourist spots (Yellowstone, etc.). At least this is the situation I have found in the south and western states, mostly (LA, TX, OK, CO, WY, NE, KS, AR). I did a 4,200 mile 30 day loop trip from Louisiana to Wyoming to do Yellowstone and the big solar eclipse last summer, of those I had advanced reservations for 9 or 10 nights (5 in and around Yellowstone, 3 nights where we camped for the eclipse, and 2 other nights that I had to be at on fixed dates, including the stop on night 1 at a small state park in east Texas with only 8 or 9 rv sites (1 of which was empty).
A couple of the nights along the way I did call to make same night reservations while on the road, though I don't think any were ever required, a couple of places were relatively full, and some first come first serve campgrounds filled up quickly around sunset, so it often paid to take ones time and get off the road well before sunset.
Having said all that some of the public, state, municipal, fairgrounds parks I stayed at on the trip last summer were nearly empty. A couple in Kansas come to mind, such as the Kansas State Fairgrounds in Hutchinson KS on a non-event weekend, 125 full hookup rv sites, and maybe 5 or 6 were occupied. Another small town public RV park behind the towns recreation building had 12 (or maybe 14) water and 30 amp electric hookup sights with dump station by the exit (what do you expect for $10 per night), had 1 other RV there on a windy Wednesday night.
2002 Safari Trek 2830 on P32 Chassis with 8.1L w/ 400 watts solar
2017 Jeep Cherokee Overland TOAD with Even Brake,
Demco Commander tow bar and Blue Ox base plate