You remind me so much of how I felt when I first got to NYU. I was so excited and proud of myself for getting in there but it all came crashing down after I started going there. I was extremely miserable but stuck with it because for some reason I had this idea in my head that leaving or transferring would make me a failure. And everyone I knew who had moved to nyc or was going to college there loved it and I felt like there was just something wrong with me.
So I've stuck it out for the past 3 years and I'm finally a senior. I always hoped at some point I would just start loving it but that never really happened. I would be so happy every time it was summer or winter break so I could go home and literally holding back tears every time I got on the plane to go back to college. Looking back I think it might have been best if I started trying to transfer schools after I didn't like my first semester. I actually think it caused me to develop depression and anxiety because I was diagnosed with it after going to nyu when I was literally the opposite, most happy and confident person before. But at the same time it has made me so much stronger and at some point you just accept that nyu can be a lonely and difficult place and learn to deal with it. I'm a lot more independent and prepared to live on my own than my brother who went to our state school.
My advice to you is to see how it is for the next few weeks. How you're feeling might just be an initial feeling that will pass, or it might not be. If it isn't, don't hesitate to try transfer to a school that may be closer to your family. At the end of the end if you're not happy it's not worth it. I mean that's the main goal in life, to be happy and enjoy it right?