Hi and welcome to the forum.
Sounds like you are planning for a big lifestyle change. Many have done this and many have attempted and then went home. You are smart for taking the time to review and weigh options. But, be careful about wearing out the guys and gals here that can really give you good advice. Like the kid at the family gathering that just keeps asking and asking questions. Don't withhold information so we can be more pinpoint with our answers. Try to focus more, too and read and look up other posts.
A couple of things that you have shared. Most National Parks won't accommodate a MH the size you are considering, period. Nor long Travel Trailers or 5 Wheels. As has been shared. Your best choices will be state parks and Corps of Engineer sites. But some of these are not open all year, as well as National Parks. As well as you will have length restrictions to traverse some Parks, Zion and Devils Tower come to mind. So maybe a little fine tuning of exactly what your goals are and work from there? Seems like you are building the boat first, before you really determine what pond you are going to be sailing in.
As for fuel, I agree with the earlier post in that what you are describing is not necessary unless you want to get fuel in one state traverse another, to save $$. So, spend $$ to transport additional fuel and sacrifice payload capacity? Remember, if you are full timing you need to carry stuff for 4 seasons, as there is no home or apartment to go back to. Where you plan on camping and traveling, you will have accessible fuel available, albeit at a higher rate.
Hopefully you will narrow down your choices and dreaming without wearing everyone out trying to help you.
b