Welcome - and have a wonderful, enjoyable holiday on this side of the pond!
Sounds like a great plan and a lot of fun. But I'll echo what someone else said and suggest you consider a small "Class C" motorhome instead. Lots of these available used. You should be able to find a good one that will carry you cross-county and back for $20,000 or less. This will leave you some cash leftover to get off the road and stay in local hotels, cabins or B&Bs when you feel like it. You could consider paying for a roadside assistance program like CoachNet to help handle any breakdowns that might occur on the road with an unfamiliar vehicle (less than $100 for a year's coverage).
Although we are fans of towable RVs, we did own a small motorhome years ago, and found it quite enjoyable in many ways. I echo the advice above, and would add that the extra savings may also help you with those city visits, and in other places where you might like to leave the camper parked and do some sightseeing by rental car. Enterprise is one rental car company that is often found in cities or other locations that are not at the airport.
The advice about buying a Class C motorhome from an RV rental company is also worth considering. In any case, I think your budget won't stretch far enough to get the kind of 5th wheel, and a truck that can pull it well, that you may need.
As far as RV Park recommendations, you're covering so much area that I'd suggest you check the site rvparkreviews.com. A lot of us use it; it's reviews of RV parks, including a few public parks (state-, federal- or locally-run), throughout the U.S. (and I believe Canada). It's a good place to start.
Good luck to you!