I live in NJ too. Have you considered Amtrak's AutoTrain? It's literally a train with storage for your car so you have it with you when you arrive. It leaves from Lorton, VA (probably 4-5 hours from you depending on where in NJ you are) and goes straight through to Sanford FL, about an hour from WDW.
I just went on Amtrak's website and plugged in random departure/return dates in August for two adults and three kids; I have no idea when you're planning to go, but I figured late August, when the rest of the nation is back in school but not NJ. If you ride coach, it'd be less than $1100, plus what it'd cost you to get to the train station and back. BUT, that includes both a sit down dinner and a walk-up continental breakfast in the dining car, and because you have your car with you, you can stock it to the roof with whatever you think you'll need for the trip, including your kid's stroller, extra diapers or pullups, non-perishable groceries to eat in the room, etc.
You're not driving straight down, so no worries about driving fatigue, overnight hotel stops, traffic (though the train could have delays, you never know), someone else is doing all that driving. You can move freely about the cars, and there is a lounge car with a bar menu (burgers and pizza type grub, not included in the dinner menu you get with your coach ticket) so you're not quite so stir crazy.
I'd also add that chances are way more than likely the $1100 for the round trip ride (and the extra 100 or so you'll spend on gas to and from the train stations) is almost definitely less than you'd spend just on airfare for 5 people. to say nothing on the extra baggage fees for your stroller (or the rental fee if you get one down at WDW), the extra baggage fees for any diapers or pullups you'd need. And if by flying you'd think you'd need a rental car, the savings become even more substantial, the tradeoff is the extra travel time, but nowadays, in a world where everyone has a library/video store/jukebox/arcade in their pocket, how much of an issue is that as long as you have those days to spend on the rails?
The train departs at 4pm (wine tasting on the train starting at 3!) and if there are no delays, will arrive in Sanford at around 9. Could take 90 minutes or so for all cars including yours to be unloaded, you should be at WDW by noon. Same departure/arrival times the opposite way so on your departure day you can still have up to 1pm or so before driving to Sanford.
You could also opt to pay extra for all or at least some of you to have a private room on the train, but that can knock the price up significantly. You can always choose a room or roomette for one parent and the 2 year old, if you think privacy would put her more at ease, and coach for the rest if the teens are cool with it.
Disclosure, our AutoTrain rides could have gone smoother, but a lot of that was our own fault and the fault of a cranky 5 month old who'd only sleep if I held her, which I did, allllll night long, both ways. We got a private car which included our own bathroom, but it was still small, and things we thought we'd need to keep our baby happy, there just wasn't room for. However, now that our kid is older (we have 2 now actually, I'd have no issue trying it again. My wife is less keen, but she hates flying even more. Probably won't do it our next trip because we won't have the time for the extra travel, it'll have to be by plane, but after that, I'd try to twist her arm.
