It's more of seeing the East coast than anything!
Then I agree with "gunning it" and getting here as fast as you can. Yes, it's tiring, but it will allow you much more time here, if that's what your goal is. It's going to take you roughly a week to get here and a week to get back, so I'd push for to take that 5th week. If you only have 4, then your timing is going to be very tight and you may spend way too much time in the truck. That said, I'd go straight to the Carolinas (Myrtle Beach and the Outer Banks), and head north from there. You can stay "coastal" all the way up the coast but I'd side-trip into Williamsburg, Yorktown and Jamestown, VA, then hit Gettysburg, PA and maybe Hershey Park, PA. Personally I'd skip VA beach but that's just my opinion. If I had to choose between Cape Cod and Maine, Maine would win. You can easily spend 2 weeks in Maine alone.
IMHO with only a month you are really pushing it. Flying here and renting an RV defeats the purpose of having the RV. I'd take the month and maybe go halfway across to slow down and enjoy the sights, and save the East Coast for when you have more time or the money/impetus to fly.
