Hi Spartana
I think it depends on how far north you plan to go. If we are talking New England/upstate New York, fall is definitely a better option. Spring can be cold, muddy, rainy, or even snowy. No bugs to speak of until June/July, though, so that's a plus. (Bugs do get very bad indeed later on . . . In certain locations the black flies will eat you alive in June/July. And then the mosquitoes come out to join the party). On the other hand, fall camping in the north east is glorious. Warm days and crisp nights. If you like the coast, the weather is still warm enough to enjoy beaches for a while too. I have a friend that goes to Cape Cod every year in late September. She loves it.
If you're going to focus mostly on the south east, I'd say either spring or fall would be fine. Probably fewer bugs in the spring. Weather seems very comparable in spring vs. fall, and either way kids will be in school so campgrounds shouldn't be very full. Probably want to avoid beach locations in March (spring break, as you mentioned), but other than that spring should be similar to fall for camping.