I really like this idea and it sounds like a nice way to take a cycle tour without having to load your panniers everyday. I agree with Stokell’s three options and unless you speak a second language or comfortable around foreign languages, you may consider another country. I think Britain is a good option for this type of tour. For Example, I would pick a small city like York, Exeter, Windsor (London) or Newcastle and enjoy the historic sights, museums, galleries, pubs and scenery in that area. Compared to America, places are relatively close together so you can make day trips to big cities or other sites. With your folding bike, you will be able to take it on the train and if it is in a bag, they will let you take in on some of the buses. Find information, contact The National Trust and English Heritage for historic sites, historic homes, gardens, etc.
If you select Canada: Quebec City, Halifax, Victoria and Niagara would be good choices as a base. June is the perfect time to visit and with the exchange rate on the dollar, Canada is affordable!