Hum is not easy to fight, and usually a system-wide issue.  Indeed, with wires and grounds going all over the place, transformers everywhere, it can be a nightmare.  Not supposed to happen, but sometimes it does.
As for the tranny, it can help if the bells connect to chassis.  Scraping a little paint off the corner foot and chassis can help with the connection.  If by touching the tranny and chassis at same time with your hands makes for a quieter signal, than that is the problem.  Sometimes that is the easy way to track down a hum.  Just use the finger test.  Skin conductance from a known ground to a questionable point, like another part of the chassis, or an RCA shield, can point to a loose connection.
If the ST35 is humming wihout an input signal, start there.  Get that machine quiet first.  Then we can start adding more to the mix.  
jh