We previously had a 36' coach and now a 40'. We camp in County, State and National parks all the time. We do fine that if you go somewhere popular, you may need to know which sites are bigger and if they're available.
On a side note.....you say you're leaving on an "epic round-the-country trip". Is this a trip of many months and several days in one spot, or travelling through several states and stopping at things like the Grand Canyon. The reason I ask..... we did a cross country trip last year with certain destinations. While driving between major attractions on our itinerary, we estimated how far we wanted to drive that day and then located a campground in that area.
If you intend to do something similar, I think it will be very difficult to locate State or County parks, know if they're any good and then hope they have an opening that will fit your coach. To me, this would take a TON of investigating and a pretty rigid itinerary where reservations would be needed.