We just got back from HHI today, but we flew. From talking to many other guests and members at the resort, it should take about 14 hours from NYC. It seems we were the only idiots to fly to Savannah and rent a minivan. We left our villa at 8:00 AM this morning and arrived in northern NJ at 4:00 PM. So, we saved about 6 hours by not driving. The cost to fly for 7 people plus the car was about $1900. Probably not worth the money to save 6 hours.
A car is truly not needed at HHI, but very helpful. Since there is no restaurant service for dinner at the resort, you must eat elsewhere or cook. There are several restaurants within a short walk of the resort. The supermarket (Piggly Wiggly) is pretty close as well. Bicycles are a must if you don't rent or take a car. Bike rentals from Disney is a great value. Al will take great care of you.
There is shuttle service to the beach house, but it stops at 4:15 PM. We took our bikes most of the time anyway. It is about 1.5 miles away. Another option, which we did not take advantage of, is something called a Private Affair, I think. A chef will cook a meal for you at the resort, and you can participate in the cooking as well.