Once you give notice, you do have to pay your rent during the time you are living there. If you don't give the notice required in your lease you are responsible for the rent. The landlord can take you to landlord/tenant court and get a judgment against you if the judge finds you are in violation of your lease.
If you would like a copy of the NJ landlord tenant handbook, contact me via Trulia or my website and I will be happy to email it over to you.
You need to check with the Lease that you signed with the Complex. Even with a "month-to-month", there may be wording regarding how much advance notice you must give (best to give it in writing so you have a record!). Likely the Lease will detail the financial penalty (if any) of not giving the required notice.
What does it say in regard to giving notice to move?
How much notice did you give?
If the lease says you need to give 60 days notice before moving, and you didn't......then yes, you should and can be responsible for the rent.
For a legal opinion.......call a local real estate attorney to clarify this.