This triggers lots of thoughts in my mind but none that have not been covered by previous replies or links in those replies. You're just gonna have to dig into it (following the previous replies) and experiment until you can remove and repeat the problem at will. When you can do that you can be pretty confident you know what to do to eliminate at least one of the problems (you may have several).
Separate wires (put some distance between them), change ends the shields are grounded on, float or bond commons, etc.
The best tools to use to fight noise issues are detailed wiring schematics and component layout drawings. Those 2 are the first steps in mitigating noise issues. Assuming they match the machine exactly you can use them to check for ground loops, star ground, proximity of components, etc. Without those the best anyone can do is make educated guesses. That route usually ends being way too much wasted effort and time to guess down problems that should have never existed in the first place. Not a game most will take time to play.