Without any question, Rodriguez Key will be your friend. We draw too much to go "inside" as well and have anchored on both sides of the Rodriguez at different times.
Rodriguez is a nice distance from Miami
Beach/Government Cut - you can make that under most conditions.
After that, look around Boot Key - you can easily enter there and get a mooring
or use one of the outside anchorage locations.
From there, Key West
is an easy passage
There are other places to go and other possibilities. We've been back behind Islamorada - that will support deeper draft vessels. There are some free moorings available around here.
is another place although it has rather poor conditions - but it's good to know about.