I’m not sure why people are thinking the Tappan Zee doesn’t have a toll. It has the worst kind of toll imaginable. It takes a picture of your license plate and mails it to whatever your address is. If you are a full timer, this can be very expensive because you never get the message in time.
However, these are the only two practical route unless you are taking a tour of upstate New York.
The Tappan Zee Drive is a bit more pleasant. Going through the Bronx on the Cross Bronx Expressway will rattle everything so hard you think you’re going to break it.