How about the train? Trains here on the West Coast seem extremely slow and horribly expensive, but I figure the East Coast should have a more developed system...
How about the bus? Greyhound?
I would drive, but specifically at night. Not only will the traffic be light, but the kids should have an easier time falling asleep. When you arrive at your destination you'll be tired as hell, but a quick 4 hour nap and later a good night's sleep will bring you up to speed quickly. When camping with my dad and his friends when I was younger, we used to do that all the time...except I'd never fall asleep.
Plus, you apparently have a wife. You drive halfway, she sleeps halfway...switch.
- LostCause