I changed my address moving into NYC but my employer did not update my withholding

I originally lived in New Jersey when I started with the company, and shortly after I moved into NYC in the middle of 2023. I changed my address as soon as I moved but unfortunately I only just realized they never updated my withholdings for 2024. I didn't catch this when I filed in 2024 because I was filing for multiple state incomes. How do I navigate remediating this? I am not in the financial situation where I can afford to foot this penalty bill. Also due to this error, can someone tell me who truthfully owes NYC the local tax, me or my employer?