We have TONS of tech shops in "central" florida, also many on here are cave instructors, and most of us all know good friends that are cave instructors. Plenty to pick from
I'd be glad to recommend my instructors if you are looking, PM me for info.
As far as what's the closest you can drive to from West Palm?...
Blue Springs State Park (Cavern) - Orange City
Paradise Springs (Cavern, cave at the very bottom, AOW allowed to dive here) - Ocala area
Devil's Den is kind of interesting (open water, but mostly cavern-ish and tight cave type swim throughs, all INSIDE a cave) - Williston
Blue Grotto is plain boring (big cavern) - Williston
Manatee Springs State Park - only about 30mins closer than the rest of the north florida caves. Has a cavern, but nothing spectacular, the cave is more interesting.
My two favorite caverns are Paradise Springs and probably Orange Grove at Peacock.
Aside from those, you just need to keep driving north to cave country (Luravilla, High Springs Live Aak areas) where everything else is.
My west palm buddy drives up often to dive with me. He usually makes it a weekend since he has to drive the extra couple hours to Orlando before we head the few hours north to the caves.