There are several caves that have been paddled through in Central America.They are mostly flat water.The Caves Branch of the Sibun ,in Belize,is a major tourist attraction ,often overrun by tubers on cruise ship junkets.It can be rafted or kayaked privately in off hours.It has multiple caves with short sections above ground in between.You must know which cave to portage.At the end there is a jungle section to the takeout that has some II's
Candelaria Caves in Guatemala is similar, but sounds better,not overrun by tourists but remote and has easy rapids in the caves.
Barton Creek Cave BZ ,( the only one of these I've done) you paddle about 4 km and back in a cave with Mayan artifacts ,towering cielings,bats ,and the usual speliothims.No rapids.
La Ventana on Rio Auguan in Honduras has been paddled through,don't know if it has rapids or has been done since the 1st D.
There are tons of caves down there ,many with artifacts.Some of them can be visited on paddling trips,like those on the upr.Chiyu,(GT)and ATM Cave on Roaring Creek(BZ)
A couple that cannot be run but are around gnarly rivers:The Cahabon River goes through Semuc Champey,a spectacular travertine bridge/staircase of pools and waterfalls above ground while the river goes subterranean .The canyon above the cave is gnar (i think still unrun) solid class V goes straight into the cave and comes out about 1 km(?)dwnstream.You can cliff jump into the pool at the bottom and swim and climb up into the cave exit,AWESOME!
There is a portage of a waterfall that drops into an abyss ,in Rio Chich(upr.Shumulja).You are running an expert run and have to repel into the cave to portage/continue down river.
You asked about US,WotSR tells of Warm Springs Creek Wyo goes through two caves.The run is V but the caves are mellow.Nearby the Popo Agie is kind of like Semuc Champey the gnarly river disappears underground.
The cone shaped islands off Phuket ,Thailand,can be paddled to and there are sea caves that you can paddle through to access the island's centers with spectacular lost world ecosystems.Really cool but probably being ruined by tourists.Then there's the Blue Grotto,way cool,but you'd likely really piss off a bunch of Italian gondoliers,if you tried to go on your own.