Unloader valve empties tank

Hello all, I use a lot of these forums for a variety of things. That being said, I'm a newbie to this forum.
I have a small 110 volt Campbell Hausfeld air compressor, about 25 years old. It's the type where the motor bolts directly to the compressor assembly, uses no belt. I've had the model number written down somewhere but can't find it this minute... goes by the name of "Big Job" or something similar.
At any rate, I've had a problem with it for a long time where the pressure builds up normally, and the pressure switch shuts off the motor and the unloader valve immediately empties the entire tank to the point that the pressure switch starts the motor again, and refills the tank... ad infinitum. The unloader never stops unloading the air except when the switch has engaged and the compressor is pumping air into the tank. I finally broke down and bought a new Superior Electric Pressure Switch LF10-1H and installed it. Straightforward, everything swapped over with no difficulties. Fired it up and it's doing the same thing. Unloader valve simply opens and drains the tank.
I'm now convinced that my problem was probably never the pressure switch and unloader valve assy to begin with. It seems to me that maybe there's a check valve that should stop the tank air from bleeding out at the unloader valve, once the unloader valve has done its job (bleed off the air pressure from the compressor head). I can't see one externally, and have resisted the temptation to disassemble everything to try and find one internally. The compressor works fine other than this bleed-off problem, so I hesitate to tear it apart.
Anyone have an idea where I might look for this or if there's some way I can add one in-line somehow?
Any ideas appreciated.
David