My brother lives in the San Diego area, and I live in Vermont, and I've driven out to visit him and his family a couple of times. It's five long days of just driving (as in maybe 10 hours behind the wheel per day) to get there on a more or less direct route. You'll be spending at least half your vacation going to and fro across this great country of ours, and quite possibly more.
I would suggest concentrating on only one destination on the east coast, and not trying to traverse the seaboard. I think New England is wonderful, but I may be slightly biased.
For seeing DC, NYC, and Boston, I think it would really make more sense to fly out and stay in a hotel. It may well be less expensive, and you'd have a lot more time to see the city or cities, and many hotels/motels are far more convenient to public transportation than most campgrounds. Driving in any of those cities (without the trailer) and attempting to find parking in them is not something I suggest if you can avoid it. All three have very good subway systems.
Things to see in and around Boston:
The Boston Science Museum is excellent, and their large van de graff generator is, well, large.
Old Ironsides is worth seeing.
The aquarium is good, though perhaps not as essential since you have Sea World rather close to home.
Plimoth Plantation is well worth visiting.
The Children's Museum is decent, though the older kids would probably find it less enthralling.
In New England, there's Boston ;)
Beyond that, Acadia national park is fantastic. There are many scenic roads through the White Mountains of New Hampshire and the Green Mountains in Vermont. (Most are amenable to RV travel, but a few would best be avoided.) The Kancamagus Highway in New Hampshire is one such road among many others.
If you like amusement parks, there are a plethora of nice older ones in the northeast: Knoebels, Kennywood, and Hersheypark in Pennsylvania; Canobie Lake park in New Hampshire; and Lake Compounce in Connecticut are some of my favorites (but there are other nice ones too).
Along the way between the coasts, I highly recommend the St Louis City Museum. At the very least, it's unique and rather difficult to describe fully.
