More facts are needed to answer your question completely. What you don't include in your posting is what agreement you made with the landlord when you moved in - it is possible to enter into an oral lease (which is valid for terms less than one year). You also said that the landlord had a lease ready on the 8th - you do not say whether you signed that lease or not. In any event, if you are planning to move out of the apartment, you should not sign any new lease. If you are unsure of your rights and responsibilities, you should consult an experienced real estate lawyer in your area.
Disclaimer: This answer is provided for informational purposes only, does not constitute legal advice, and does not create an attorney-client relationship. Actual legal advice can only be provided after completing a comprehensive consultation in which all of the relevant facts are discussed and reviewed.
I would have to review the lease to be sure. Does the lease ending February 29, 2012 provide for renewals. It is quite possible that by accepting March rent you are now a month-to-month tenants and they can evict you by giving 15 days notice.
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No, you don't have to sign the new lease. You may have had a responsibility to send a non-renewal letter to the landlord/management. Read your written lease agreement and consider seeing a local landlord/tenant lawyer to go over it with you. It is possible to automatically renew a lease agreement without signing anything. You should put in writing immediately your intention to move out and on what day and time. Send the letter certified mail to management.
When moving out, take pictures of the premises very carefully and take notes. Then, turn over the keys to management and give them your forwarding address.
Always take pictures upon moving in a new place and when moving out. This is your protection that the landlord won't be able to take advantage of you and blame you for damages that are not your fault.
Also, read carefully to see if you have to send a letter stating that you are not renewing the lease and mark it on the calendar as to what date you must send the letter. Good luck!