Good luck with your adventure. We're ex-pats from Leyland (Lancs) and have lived in the US since mid-1968. Most of the time has been in Washington State, where the climate is similar to the SW UK or NW France.
We spent 3 years in Virginia and found that whole SE area to be way too hot and sticky. My first day leaving home to start work at NASA, it was 81F, foggy and a thunderstorm was going on. That was at 06:45 in early June.
I got my leg pulled about having lived in the Seattle area "where it rains a lot". My response was, "Yeah it rains a lot, but it doesn't rain much", In VA, we'd get summer afternoon thunderstorms that would dump 3" of rain in 45 minutes. In WA it can rain for 24 hours straight and we get about 3/4".
I hope you can adapt to the climate on the East coast quickly. Many of the places you're planning to visit have that same summer humidity issue. With luck you'll adapt well and maybe make it into our neck of the woods on one of your later visits. We have real mountains here - those with snow on the upper sections all year. Mount Baker, about 75 miles east of here is around 9000'. Mount Rainier, south of Seattle is over 14,000'. The San Juan Islands, where we live are spectacular. The Oregon Coast is a wonderful trip - we've done it 3 times.
Send me a PM if you're interested in getting tourist info and I'll pass on some web-sites, etc.
Unfortunately, we'll be "former RVers" soon. We're putting our MH on the market this week. On the US equivalent of an OAP and a smallish corporate pension, we just can't justify spending big chunks of our retirement savings running it and paying CG fees. Both gasoline and CG fees have doubled since we started 10 years ago.