Have you considered the excess baggage fees which you will incur to have so many bags? It may be less expensive to ship some of the items ahead rather than try to fly with all of it.

Separate from that, a 15-passenger van is probably the best way to move that much stuff and people, particularly if they can take the back bench out for more baggage space. I would not expect to find one of those cheap, especially with a driver. But you may be able to get one from a rental car company at EWR with a one-way rental returning at JFK.

Porters will likely be available, depending on the terminal. Assuming you drive yourself rather than having someone drive you I'd drive straight to the terminal and unload there before returning the car. Then have someone wait with the stuff & the 2 kids while the other adult deals with the car return. You will not be able to leave the car/van on the curb while handling all the check-in formalities so leave plenty of time for all of this; I'd say an extra hour on top of whenever you would normally arrive with bags to check.