I would avoid traveling on I-95 between Boston and Washington, DC as much as practical. In particular, taking it through DC, Baltimore, and NYC is no fun at all. Depending on where you're starting from in North Carolina, I-77 to I-81 might be a good alternative, or perhaps the Chesapeake Bay bridge/tunnel (US 13).
For visiting NYC, probably the most convenient place to camp is Liberty Harbor across the water in New Jersey. Having the dog may well complicate things somewhat there; they are not permitted on the subway unless in a carrier of some sort, and of course are not welcome at all in many museums, etc.
Be aware that RVs (or, more specifically, propane) is not allowed in NYC and Boston tunnels, and plan routes accordingly. Also be aware that RVs are not permitted on parkways around New York and New England, with very, very few exceptions; the Garden State Parkway in New Jersey is one important exception.
If you like amusement parks, Canboie Lake Park in Salem, NH is a very nice independently-owned traditional park. They don't nickle and dime you for things like parking, either, which is a refreshing change from many places.