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Do I need to file a state return for Florida and do I need to fill out a DR-405."
You do not need to file a return with the State of Florida since Florida does not have a state individual income tax.
You also do not need to file a DR-405 for the current year since the report would be for any tangible personal property used in connection with your rental in place as of January 1, 2017 and, ostensibly, you sold the home in May of 2016.