J.B.
ETA: great suggestion from mynewnickname to rent a house for a year before buying. Did you ever read Superfudge by Judy Blume? Remember how they rented a house in NJ for a while and then decided to move back to the city? You might not be able to find a rental in the exact place that you would buy, but it would be a great compromise to make sure it's really what you all want before investing in buying a house. There are some communities in NJ that should have a robust housing rental market to accommodate executives who transfer to work in NYC and need to rent before moving their families here.
Original: If driving is your only hang up, that's a problem that you can easily solve and are taking steps towards it. You'll be fine driving, and it's really a skill that everyone should have. Some of my colleagues travel to visit clients in places where there is no public transportation or even a reliable taxi (or Uber) service in place and the only option is to rent a car at the airport and drive to the client's work site. Having your license and knowing how to drive will literally open up the whole world to you - imagine being able to take a road trip all on your own! Imagine deciding one day that you want to go to the beach, or go camping or apple picking or on a bike trail or something else that is a pain to do in the city and having the convenience of hopping into your car with all the stuff that you need and just going. My sister lives in Brooklyn and has a car but a huge treat for her is to come home for Christmas and be able to go to a full-size grocery store or Target, where everything is in one place, not having to go to the green grocer, then to the butcher, then to the fish monger, etc. While that's fine for every day shopping needs, when she has a party or is hosting a dinner, it's a pain. I can't imagine my kids playing the sports they do without a car - hockey bags, lacrosse bags, etc. would be a giant pain on a subway or bus, and there is literally no way to get to the rinks or fields, especially for tournaments, without driving. Sometimes my kids will be a different tournaments in different states on the same weekend, so I'll have to drive to one and their dad will have to drive to the other.
Don't let this get in the way of what could be a great move for your family. I loved living in Boston when I was in college and just out of school, but once I had kids, it became painfully obvious that life was more affordable and easier out in the suburbs. I don't have a single friend or colleague who has kids who still lives in the city. I still commute into the city for work every day, and might move back when my kids are grown, but for raising kids, the move out to suburbia has been great. My parents were both raised in Boston and moved out when they got married and have never looked back.