We're also from Long Island and have made the trip at least 10 times.
We find that a 3 AM departure will get you just south of DC by 8:30 AM. You will have no problems in NY. We use the Southern State/Belt/Verrazzano/West Shore Expwy/Outerbridge to NJ Tpke.
If you go west on the beltway rather than east, you will avoid the infamous Wilson bridge and the exit onto I-95 from the beltway is easier.
Use I-295 to bypass Richmond. Keep close to the speed limit there are unmarked cars, but the traffic is minimal.
We usually stop in Lumberton, NC, which is 20 miles north of the border with SC. There's a Comfort Suites there which is decent. Tons of fast food, too.
From Lumberton it is about 8-9 hours to Disney depending on traffic and weather. If you leave at 6 AM, you'll be there by 2-3 PM. If you don't mind driving in pitch dark and occasionally, fog, you can leave at 4 AM for a noon arrival.
Going home, you may find that you will hit DC near rush hour. If that will happen, we get off I-95, take 301 to the Potomac River Bridge, to Route 3 to I-97 to I-695 to the Key bridge and rejoin I-95 north of Baltimore. This route has lights, but is usually much better than sitting on I-95 going nowhere.
Have a great trip. The biggest mistake we made was trying to do too much. It is not a bad idea to just lay by the pool after arrival, relax and then wake up refreshed and ready to go the next day. Especially with little kids some down time in the pool after arrival provides physical activity without the overstimulation of immediately jumping into the parks which can result in exhausted overtired kids and short tempers. You have to know yourself and your kids and make the choices you think will work best for you.