Yes, it could be done but it's going to take a long time to see the country.
You will not see the whole state by staying one month in it and each state and each area in the state has different terrain and different attractions.
If you move to another part of that state you still won't see it by day trips. OK... move to another part for a month and call that state quits. Now you've used 3 months and you want to full-time for only a year? You also need to think about temperatures. You don't want to be in the southern states in summer and not in the northern states in winter.
If your goal is to see the country you'll have to move more often or else plan on full-timing a longer period of time in order to see it all. If you need to cut down on cost then learn how to dry camp (no hookups) and use public lands to do so. Stay in a RV park every so often to do a thorough cleaning, long hot unlimited showers and regroup.
Join the Escapees RV Club and use their home parks and co-ops. At the present time monthly fees can be as low as $265 and the highest $450/m. Weekly would be $90 - $166; some including electric.
We full-timed for 16 years, traveling constantly - including a full summer in Alaska, and still didn't see it all. Typically we'd choose a state or two for a season and thoroughly cover those states, moving every week or so. We loved boondocking (dry camping). This is not to be confused with pavement overnighting such as on WalMart's asphalt. We stayed in very scenic areas with rivers, mountains and lakes - in the boonies.
Good luck with your plans!
