Ok Annie - here's my 2c. I think you should re-consider flying. Unless your mom's fear is really high, it would probably be the most comfortable and economical option for you. On the surface, it sounds more expensive, but, as compared to:
the train - you won't need the "free" meals they provide, since you will be flying for only 3 hours, you'll save on the $800 (is that what you said?) hotel room in NY (is that both ways?), you'll "save" on the broadway tickets (I know you really wanted to do this, but if you are flying, there's no option anyway, so you won't be tempted).
driving - consider the hotel cost on the way there and back, that gas is nearing $4 US per gallon and it could be a very expensive trip.
Also consider the small price you will pay for the convenience of a short trip (plane ride) vs. the train or a car. Based on how you describe your condition, it seems you wouldn't be too comfortable being "cooped up" in the car for several hours a day either.
If you fly, yes, you get to the airport 2 hours early, which is early early morning, based on what you said above, but some people (including myself) can sleep on the flight. Also, you could build in some rest time as soon as you get to the resort.
You would still not have a car at WDW, but you could consider that taxi to the shopping center (or consider a 1-day car rental from the agency in the Dolphin, which might be less expensive than a round-trip cab ride).
You do still have similar luggage issues (though not the same - as you CAN check bags) to the train, with flying, but consider that if you drove, you might pack more luggage anyway because you're adding days to the front and back of the trip and need more clothes, etc. So you may still be limited on space in your car.
Anyway, just my thoughts. I hope this doesn't complicate the situation for you. And if Mom is uncomfortable flying, I guess you can go ahead and rule that out anyway.
I'm sorry you're having so much trouble with this decision, but soon (within days) it will be all worked out and you will feel that sense of relief again.
We considered driving, and we love a good roadtrip, but we ultimately decided to fly for the convenience of it.