Just a minor point - a buzz is 120Hz. 60Hz noise will typically be a soft tone associated with the radiated magnetic field of nearby transformers. I don't see anything in your description that completely rules out a ground loop problem. A simple test is to use a cheater plug on the FPIII and see if it changes anything. It's possible that 13+ year old electrolytic caps are causing the problem, or bad tube pin connections, or oxidized switch contacts, or oxidized attenuator contacts...
So my suggestion is eliminate the easy to test stuff first. Along with the cheater plug try cleaning tube pins, sockets and switch contacts as well as the rca jacks. Only after all that consider replacing components.