OP: The best thing for you to do is make a list of the parks you want to visit and check their websites for more accurate information.
With our 40' motorhome our first choice is always a public park - national, state, COE, national forest, county and city.
We do our research and have never been turned down at a park of our choice. Plus, we don't make reservations. We arrive early morning, staying the previous night closeby and sites are easily secured. Having volunteered in national and state parks we know that they don't have reservations for every single site and some campgrounds don't even accept reservations.
Granted, we mainly stay west of the Mississippi nowdays. If your plans are to stay solely in the eastern states then you might have problems. We have stayed multiple times in Yellowstone, Grand Tetons, Glacier, Zion, Bryce, Canyonlands, Denali's Teklanika and Riley Creek, Jasper, Baniff and Rocky Mountain just to name a few.
No, not every site will accommodate you but there are plenty that will. Motorhomes like ours don't bend in the middle like a 5th wheel but still, we have no problems.