Can I still collect unemployment benefits from New York State and get a part time job in New Jersey where I live????
That would depend on your part-time job.
This is what the NY website says ..
If a claimant works less than four days in a week and earns $405 or less, the claimant may receive partial benefits. Receiving partial benefits extends the length of time benefits can be collected until the claimant receives his/her maximum benefit amount or until the benefit year ends.