Does your lease address early termination and default? That is where you can find the answer.
I provide a free 15 minute telephone consult for security deposit claims and eviction defense. No attorney-client relationship is created by answering questions in this public forum. If you wish to create an attorney-client relationship, you must contact me directly and sign a representation agreement. Answers are provided based on general ideas and an answer specific to your situation would require a review of all documents.
"IT DEPENDS." In order to answer your question, we would have to review all written lease agreements and addenda in conjunction with Florida Statutes Chapter 83, section 83.595, as well as run the numbers to find the correct balance on your account.
In some cases, the property owner can charge "double rent," "liquidated damages," "early termination fee," or the like.
If it is contemplated and put into writing, then it is very clear that the LL can charge both parties for the same time.
If it was not contemplated by the parties and put into writing, then an analysis of the terms and conditions of the lease and a detailed accounting of the giving of the notice that you gave, the date you moved out, the date the new TT moved in, and all of the other relevant facts would have to follow.
I think if you didn't have a clause for liquidated damages / early termination fee in the lease agreement or any modifications, you may prevail, but I can't say for sure without hearing more details, so see an attorney.
If you are legally owed money back, then the owner will have to apply that as a credit on your balance. The owner has to send you a letter within 30 days after you vacate if they intend to make a claim on your deposit. After you receive it, you must reply back immediately by certified mail that you dispute the claim(s) and that you demand all of your money back (whether paid as deposit or as payments made during the tenancy which should be refunded as a credit).
(I say "immediately" to make sure you meet the time frame, but, actually, the former LL has to receive it within fifteen days after you receive his letter.)
If there is no deposit, then the owner should still send you a statement of your balance. If they don't do that within 30 days, then you can send a demand letter certified mail asking for a return of all moneys paid that should be refunded.
If you have to sue in small claims on your own, it will cost starting at $55.00 for the lawsuit and may go up to $300.00 or $400.00, $10.00 for the summons, starting at $40.00 for the service of process. If you have to hire an attorney, you'll have to pay a retainer up front and then for any overage. If you prevail, you should be able to have the judgment for the money you are owed plus the court costs and attorney fees. Then, if you are lucky, the LL will pay.
A letter from an attorney may solve the issue.