I think a lot of it depends on the time of year and the day of the week.
We took a ten week / ~11,000 mile trip this time last fall and I think the only reservations we had were actually on the beach in Port San Luis, CA. For the rest of the trip, we usually just made a phone call early in the afternoon to book a spot. Even then, the only reason we called ahead was to save a spot since we're oversized (45' 5er).
The only time we had to do our Las Posadas imitation going from park to park begging for room at the "inn" was in Bishop, CA on our return. It turned out they were having a big car show and the town was pretty packed. We found a spot eventually though.
If you're traveling off-peak though, it shouldn't be a problem to vagabond around the country with a loose itinerary. And hopefully you can boondock if there's truly no room at the inn.