Since you rode at Caravelle Ranch WMA I assume you are in NE Florida. I am over in Tallahassee and haven;t found that any more than normal mountain bike tires are needed for the trails here. Even my bike with 2.1" tires seems okay on our sandier trails IME. My bike with wider 2.4" tubeless tires running lower pressure may inspire a bit more confidence, but the 2.1" with tubes are okay.

There may be more sand where you are, but I bet it is a similar range of conditions and you can choose to ride in any of them. I doubt it is worth going to a fat bike unless that is just something you really want to do.