Here is another test you can run: connect a 6' wire to the ground side of the BNC connector on the radio, with nothing to the center pin. Extend it as you would a jumper cable. Do you get any interference?
Very little. So better shielding might help, I suppose. The problem seems to be frequency dependent -- less on 17m than 40m.
Using a T-R switch...
What type of T-R switch? Not a relay I presume, because it couldn't operate fast enough. So an electronic T-R switch?
Two kinds, actually. My K3 has a transverter adapter, which allows me to get receiver RF off the antenna. This is the arrangement I tried first. Then a DX Engineering RTR-1A. Both use relays, and both exhibit the QRM.
A fast switch acts as a modulator, so it may well superimpose your signal on the others it is hearing because of that.
Quite possibly, or on top of the normal background noise.