60Hz hum with piezo disc
A piezo transducer produces no voltage when truly "at rest".  Perhaps there are small vibrations where you are testing, like a fan running in equipment on the same bench.  Or, you are picking up power line or another signal capacitively.  Piezos have very high impedance, so capacitive pickup is quite likely.

To test this properly, shield the lines and mechanically isolate the piezo and the wires going to it.  Set it on a folded up towel or similar, with a grounded piece of aluminum foil underneath the towel.  If that significantly reduces the signal you are seeing, then you were clearly getting mechanical vibrations or capacitive pickup previously.

You really should use a shielded cable for the connection, like something intended for sensitive audio signals.