Connect pressure to the purge port on the dryer.
The air exits the dryer thru a check valve to the wet tank. If that check were leaking your shop air test should show that. The system feeds air from the wet tank to the governor the governor monitors this air to control its cycling, and also feeds this air back to the dryer purge port When it unloads at 125PSI.
Not sure if the governor will unload without the compressor/engine running, but I suspect it probably will, but you need to apply enough air pressure to the system to exceed its cutoff point, which should be ~125 PSI. You would know its happening as it sends air back on that purge line to the dryer purge port.
with the system at 120, I suspect you could disconnect the dryer purge line, apply 120PSI to the dryer purge port and look for leaks, and also apply 120 PSi to that purge line back toward the governor and look for leaks along that line.
a leak on that circuit would explain your symptoms, static system holds pressure well. When running, system fills, governor unloads, dryer purges and the leak quickly drops wet tank pressure to the cut-in point, leak stops when the purge line is depressurized by the governor, and the compressor quickly refills the system only to re-apply the leak and start the 2min egg timer all over again…
Arn’t air systems cool