The answer is:
You can operate a golf cart on (or across) a public road ONLY if the appropriate governmental agency approves it.
County Road? Contact County government.
State Road? Contact State government.
An excerpt from the State Law regarding golf carts on public roads reads:
The 2013 Florida Statutes
Title XXIII
MOTOR VEHICLES
Chapter 316
STATE UNIFORM TRAFFIC CONTROL
View Entire Chapter
316.212 Operation of golf carts on certain roadways.—The operation of a golf cart upon the public roads or streets of this state is prohibited except as provided herein:
(1) A golf cart may be operated only upon a county road that has been designated by a county, or a municipal street that has been designated by a municipality, for use by golf carts. Prior to making such a designation, the responsible local governmental entity must first determine that golf carts may safely travel on or cross the public road or street, considering factors including the speed, volume, and character of motor vehicle traffic using the road or street. Upon a determination that golf carts may be safely operated on a designated road or street, the responsible governmental entity shall post appropriate signs to indicate that such operation is allowed.
The full text is at:
Statutes & Constitution :View Statutes : Online Sunshine