You can do Airbnb for longer periods. Many places have discounts for long stays, but if there's already shorter bookings mid-stay you won't find those long term listed.
There's also VRBO.
One thing I don't like about Airbnb is the fees... you're going to pay at least 12% extra to Airbnb. Another 3% if paying in another currency. And then Airbnb charges the hosts too. I've seen 3% for service fees, but a place I visited in Indonesia claimed they were charging them 18%!
Craigslist will likely be your cheapest option. You might find some deals there but probably get the best deals if you don't book too far ahead. So maybe Airbnb for a few days while you look for a place.
The Keys in those months is not cheap. Several years ago, the cheapest Airbnb I could find was $60 a night + fees for early November. I also stayed a couple days at this fisherman's motel in Islamorada. It's just a block from my favorite bar Lorlei, great happy hour, and they do weekly rates. The motel is OK - clean, wifi, cable TV, nice A/C.
http://www.keylantern.com/rates.htmLooks like if you do the weekly rate, the nightly rate for 1 bed is $55.71+ taxes. Maybe can negotiate better if staying longer.
Another option is camping, but that's still going to be $40 per night or more.