So how many ohms resistance do you measure between the chassis of the T-3 and the metal body of the probe.... it should be very close to zero ohms if the shield braid is connected correctly and has continuity. Since you can quiet the hum by touching the ground side of the cable (where?) , tells me you still have a ground connection problem.
Does the hum change if you reverse the power plug in the AC socket?
Ohm out the diode as well, if you have the probe body opened again.
jr