You should expect the landlord to initiate eviction proceedings if they want you to move out.
This is not legal advice nor does it create an attorney-client relationship. This is for education and informational purposes only. It is always recommended that you contact an attorney with any concerns as each individual case is unique.
Your tenancy is not subject to rent regulation if your lease lapsed and no renewal lease was offered. The fate of a New York tenant without a lease, is legally subject to the whim of the landlord, and a month to month tenancy, may come to an end in NYC as long as the landlord properly serves a written thirty day tenancy termination notice.
Should the landlord terminate a month to month tenancy, a tenant defending the holdover proceeding who is actively looking for a replacement home, may turn to the Housing Court judge, and request an enhanced stay of eviction to continue to look for a new home. The judge can do so as long as you continue to pay the landlord (if directed by the judge) and there is cap to the amount of extra time a tenant may gain from the judge.
Settling a case in Housing Court without an attorney carries special risk and so you should consider seeking assistance from an attorney.
The answer provided to you is in the nature of general information. The general proposition being that you should try to avoid a bad outcome if you can.
Please note that in addition to terminating a month-to-month tenancy, the landlord can bring a holdover proceeding immediately upon the end of your lease term if she does not accept any rent from you for the month after the lease terminates.
The above constitutes general information only and should not be considered legal advice.