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Unfortunately, your landlord has a right to unilaterally terminate your month to month tenancy with only 15 days notice (and they can do so without providing any reason).
See FL Statute: http://www.leg.state.fl.us/Statutes/index.cfm?App_mode=Display_Statute&Search_String=&URL=0000-0099/0083/Sections/0083.57.html
There is no provision of the ADA or other code that entitles you to additional notice.
You can try to get your landlord to provide you with additional time (the most convincing requests are those that show the landlord you are making arrangements for a new place and will be out within a specific time frame (such as "I will be vacating within 25 days, but my new residence will not be available, please consider extending the termination of my lease until that day so that we can avoid the time and expense of an unnecessary unlawful detainer").
However, if you fail to vacate within the 15 days, and have not yet convinced the landlord to provide you with additional time to move, your landlord cannot simply lock you out. To remove you from the property, they must file what is called an "unlawful detainer" action against you (a court action), if you appear in court and defend the matter (which I strongly encourage), it will delay the eviction (sometimes for a couple of weeks, up to a month or month and a half - it depends on how backlogged your court is). After the landlord gets their judgment of possession, the county sheriff will then forcibly remove you from the property (so it is in your best interest to be actively moving all of your property throughout this time frame).
If you need assistance finding new housing, and help moving, contact your local (county) housing authority, let them know you are facing imminent eviction and you need placement and housing assistance. They will place you at the top of the list, and will put you in touch with local assistance.