Classic appearance of common mode noise. Connect the ground near where the signals come from, then make a common mode transformer by carrying one extra ground wire with the bundle of wires, through a suitable core, back to the main ground pin where the power wires connect. The ground loop voltage through the extra wire (which would ordinarily be a shorted turn, but that ignores the fact that ground isn't, in fact, ground, everywhere on the board) induces a similar voltage in all the other signals, keeping them quiet(er).
Maybe.
Use a few turns on a large ferrite, usually a clip-on cable clamp style, but a toroid (ferrite, not powdered iron -- if it's not dark gray, and you don't know what it came from, assume it's not) will do fine, just a little inconvenient to loop your wires through.
Tim