Do you have a real estate license? If so, any real estate brokerage does leasing, especially in the larger cities in Florida because there are so many condo buildings.
As far as the property management, I believe you have to contact all of the big companies and they my start you at the smaller buildings but it seems that it is a great time to find work in that field. But possibly a lot of realtors decided to go to management so that is why there are not many jobs.
South Florida Brokers