Dairy free in NYC

I am an international student that just moved to NYC to study.
I am severely lactose intolerant, I will get really sick and get dermatitis (not allergic, I can have dairy free milk), and the pills don't really work. I could try taking more, but I don't really want to risk it.
I thought that I would not have issues in NYC, because it's a big city, but it's bad. I can find dairy free options in the grocery store, but if I go to a restaurant and say that I'm lactose intolerant and can't have dairy, they look at me like I'm crazy and ask what dairy is. I had to explain to my school that I am not vegan so I can have meat in my meal plan, but not dairy, and they where confused by the term "animal protein".
What the fuck guys?
Am I going crazy? Is being lactose intollerant that uncommon? I am Italian, and in Italy we even have dairy free pizza, I always find options at restaurants, and people understand what dairy is.