What date are they saying they need it by?
On my lease I usually see what happens if I don’t renew it for the year, example they will triple my rent if I live month to month, can you check if they have that option if not you will need to ask
Legally they have to send it out a certain amount of days before the end of the lease when there's a rent increase. It typically says when you need to sign and return it by on the paperwork, which is usually the month before the end of your lease. Ask about penalties for ending the lease early if you can't find a place soon and need to renew. For example not getting your deposit back or having to pay an extra month.
You lease will include a clause regarding the consequences of a month to month tenancy and if it’s even possible. They may not allow it and may institute eviction proceedings. The fact that you are in NYC provides protections. Call 8338750865 the tenant helpline and they will answer your questions or refer to the housing court attorneys in your borough to answer your questions.
I suggest you tell them the truth and that maybe they can give you a month to month instead so you're not rushing to get out or stuck for another term you don't want