Covered slips in the Keys are few and far between and usually spoken for. At 36' there may be a few inside forklift storage options for you, but again they are usually at capacity.
Tavernier Creek Marina can handle up to 36' and 12,000lbs
Tavernier Creek Marina
Duck Key Marina can handle to 22,000lbs
Keys Boat Works Marathon
Boat Repair Services & Products
Oceans Edge formerly Oceanside Marina Stock Island (Key West)
There are several others, Plantation Boat Mart, Max's, Caribee, Bud n Mary's, Boat House, Marathon Boat Yard, Keys Sea Center, Garrison Bight, plus a few others. I left them out because I believe you either too big or I doubt they have room.
I'm pretty sure the only covered in the water storage in the Keys is at Key's Boat Works in Marathon, maybe Marathon Boat Yard as well.
Note that Marina's in Marathon and South may have been impacted by Irma.
You would do much better in Miami and still be only a few hours from the Keys, or opt for in the water or on the hill outside storage