Some states issue an EZ Pass to non-residents. The fees by license plate photo result in higher toll fees plus a service fee. All that can add up quickly. For toll road planning you can explore the tradeoffs in extra gas vs shorter toll roads:
https://tollguru.com/
Nonetheless, I would swing widely south of Chicago to I-70 and back-up again to NJ, due to the awful toll roads and traffic in IL.

Your plan almost begs a fly-rail-drive combination. As your plans begs a lot or road time in a short amount of time for the distance to driven, it might be a more enjoyable option to explore alternative options.

It is easy to train from downtown New York City (or the New Jersey suburbs) to downtown D.C. Stopping in Philadelphia along the way is also an option to see some Independence sights. From D.C. you can fly or train to FL (fly from DCA to Jacksonville, FL or Orlando, FL; training has several nearby city options from DC to DeLand-area).

If you are driving, riding the AutoTrain from just south of D.C. (Lorton, VA) to just north or Orlando (Sanford) is an alternative. You pay for the vehicle and the passengers. It is basically a noon to noon 24-hr trip (the train leaves at 3:45p and arrives at 9:00a, but your car needs to be check-in and loading and unloading; dinner and continental breakfast are included).

I love Road Trips. But I have the opposite feeling regarding traffic jams. And NYC and DC are traffic bogs (actually the greater NYC area plus greater DC from Aberdeen, MD to Fredericksburg or Richmond, VA). It is even worse if you run into all weekend beach-related traffic jams at several points along I-95.

NYC and D.C. are great cities. Both have superb museums--NYC mostly paid admission and DC almost all free (Smithsonians). NYC has Broadway, Central Park, deli's; D.C. has museums, monuments, and more.

If you choose a road trip it you might want to start plotting plans on a spreadsheet. My take is that 400 miles can be a long day logistically with 3 teenagers in-tow.