Thanks, Reb!
According to this, from the site you've got linked, it looks like I'm off the hook:
You do not need a license to hunt hogs on private property with the landowner's permission.
If you are hunting hogs on public lands, a license is required. Permits and license requirements for hunting on public wildlife management areas (WMAs) are spelled out in the brochure published for each area. You can access a copy of these brochures on the Florida Fish and Wildlife Conservation Commission's web at:
http://myfwc.com/hunting/wma/2005-06-Regs-Maps.htm.
Hunters wanting to use WMAs during quota hunt periods must submit applications unless license exemptions are allowed. There are several types of quota hunt applications. The two most commonly used are regular quota hunt and special quota hunt applications. To learn more about the Quota Hunt Program go to:
http://myfwc.com/hunting/quota_hunt/default.html.
To find out which quota hunts allow hunting license exemptions go to:
http://myfwc.com/hunting/quota_hunt/default.html#worksheets and select a Quota Hunt Application Worksheet. The hunts that do not allow exemptions (require permits) are printed in bold.
Exempt hunters may apply for areas with no exemptions and may apply for areas with exemptions only during phase I to be included in the antlerless deer drawing.