In many analog pro audio applications, it's actually recommended that a shield be connected at one side only, for this reason. By convention but not necessarily necessity, the bond is typically kept at the receiving end, as that's almost always a device with a grounded power cord (such as a mixer). Many DI boxes feature a ground lift switch as a convenient way to achieve this. But you wouldn't want to disconnect it at both ends, as then the shield has no effect at all.

Anyway, if you had problems with your unshielded cables that would be solved by a shield, but your shielded cables caused a different problem due to the bond at both ends, this technique of using shielded cables but severing the shield at one end of them would get you the best of both worlds.

Ruled out the monitor(s)? There's been cases where they've backfed power, and they certainly backfeed EMI as well. And, it could also be tied to FPS- assuming gsync/free sync.

If you have a multimeter its probably worth double checking if the case is low resistance grounded to the end of the cord. I'm assuming you have checked already, but as a shock hazard it bares repeating.