You do not have to move.
You tried to tender rent for June and the landlord refused and returned it.
The LL may have lied to the new owner that there was no tenant in the property.
The new owner has to honor your month-to-month lease.
You should plan to meet with the new owner on June 30th to give keys and surrender possession.
Remove all of your items, clean thoroughly, take lots of pictures, then turn over keys. You might let them know that you took lots of pictures in case there is a dispute later.
Find out if the former owner is turning over your security deposit to the new owner.
Let them know your forwarding mailing address.
Send a demand letter if they don't write you within 30 days after you vacate either returning your deposit to you or writing you a letter making a claim against the deposit. Send your letters to them certified mail.
If you are a month to month tenant, your landlord must give you at least 30 days prior notice. If your landlord told you less than 30 days ago that he wants you out by June 6, this is improper notice. As for your security deposit, upon inspection of your apartment, if there are no repairs that need to be made, not including normal wear and tear, then your landlord must return your security deposit.
This response will not create an attorney-client relationship.
Fla Stat 83.57 specifically identifies the time a landlord must provide you with notice. Specifically in your case the statute reads as follows, "When the tenancy is from month to month, by giving not less than 15 days’ notice prior to the end of any monthly period;" This is the notice that the Landlord must give you to TERMINATE your month to month lease. After that if he wishes to remove you he must file a judicial action in county court. In such action you may raise the defense that such notice was ineffective because it was not timely even. In regards to the security deposit the Plaintiff must provide you with a proper notice for any claims that they wish to make but such can only be for damages that exceed normal wear and tear.