The part to leave is the tourist part and go seek out treasures outside Manhattan. Go to Coney Island on a weekday when there's no one there. Take the Staten Island Ferry (it's free) there and back when the weather is nice enough. Go to Brooklyn and visit the Brooklyn Museum. There's a lot to see in New York and none of it involves Times Square or visiting the Statue of Liberty.
New York City yes, but there is an entire state with beautiful outdoors, farmers markets, rural and suburban cities, decent universities, tons of craft beer, art, and... Snow. I feel bad for Buffalo.
Point is after moving up here I get why natives get annoyed over the whole New York State vs City thing. Cost of living is definitely not that high state wide either. It isn't perfect, but it is home.
The rent only ever increases. Demand was too high for too long. We were paying 2k for a 2 bedroom on SI. We now rent a 2 BR house for nearly half that upstate.