Unemployment is not a needs based program. It's more like insurance - if you qualify you get it; if you don't qualify you don't. There are very, very few reasons under which you can quit your job and collect unemployment and with only one exception I can think of, those reasons are all job related. (The exception, which applies in only a very few states, is if you quit to follow a spouse who has been transferred beyond commuting distance.) You had a good reason to move, but since your reason was not job-related (you didn't quit because of harassment, or not being paid) you do not qualify for UI. The employer's support does not change the law.

Why would NY allow you UI? No one paid into the NY unemployment system on your behalf.