Digital Multimeter "Continuity test" only one single beep
A short burst of beeping from a continuity tester generally means that there is momentary current flow in the circuit from the tester until some capacitance gets charged up and the current flow stops. 

On the other hand the continuous tone indicates an actual short and the current from the tester flows as long as the probes are attached to the circuit under test.

So if you are seeing your "bad unit" show the continuous beeping it probably means that there is a short circuit.