2001 Oldsmobile Alero fuel tank

There's definitely not an overflow anywhere. All fuel systems are sealed to prevent evaporation, and since 1996, all cars have a method of detecting even very small leaks.

It's fairly common for gas tanks to rust out, but given your observation there has to be gas in the filler tube before the leaking starts, it's more likely there is a rusted hose clamp on the filler tube or something like that in that area. You'll need to have it inspected by a mechanic, but my suspicion is the repair is going to be a lot less costly than for replacing the tank.