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You are responsible for the lease payments unless and until the landlord rents the place again. And in every state that I know of, the landlord is required to "mitigate", which in this case means that he or she has to try to rent the place right away to someone else. Let's say you move out on February 3, and the landlord puts an ad in the paper, shows it to a few people throughout February, and rents it for April 1. You would probably be responsible for February and March at worst. At best, XXXXX XXXXX to February 3. But, chances are it could not really get rented in the middle of February.
The landlord cannot: "double dip"--charge you for the remaining months of your lease and also collect rent from a new tenant;
Nor can the landlord let the place set empty until your lease is up and expect you to pay.
Explain the law to them and tell them you will be fair, if they are.