"Resistor didn't help.
Really I don't think it's an issue with the wiring as much as the antenna and/or circuit board itself throwing out a signal that the cable is picking up."
There are, as I see it, 2 possible causes - The first is unlikely.. but IF you have no wiring / grounding problems.. well .. perhaps.. IF Modulated RF is being transmitted from the board, and IF this is being demodulated in the cable / connector / input circuitry (any non-linear component, or even oxide on the connector, might do this) then any attempt to break the signal with a switch, and loading the input with a resistor, may not work / may have unexpected results... Disprove this by trying a different amplifier and cable..
Otherwise, it is a less esoteric problem..
Ok - lets go back to basics.. In order for a "cable to pick up" signal, that signal must, somehow, be coupled to the cable...
Mechanisms for signal coupling are: Resistive, capacitive, inductive.. more than one of these can (and usually are) be acting at the same time. These are defined by their impedence (Z)
Now.. The amplitude of the coupling will depend on the source Z (sZ), coupling Z (cZ), and the load Z (lZ). The Output level (at the cable / input to amplifier.. ie, signal seen across lZ)will be = 1 / (Output Level * (sZ+cZ)/lZ)..
So, lets take an example.. 1V from a low Z output (say 100 ohms) via 100k coupling (cZ=100k)into a 10k load (lZ=10k) gives an input of 1/(100/10) = 1/10th of the signal from the source..
Into a 10k load resistor, at audio frequencies, extremely large capacitive coupling would be required to inject signal into the cable.. Inductive coupling is even less likely..
There is something basic wrong with what you are doing... I am starting to think that you have a grounding problem... If your cable grounding is bust, then connecting a resistor between signal and ground will have no effect.. There must be some problem of this kind... Audio frequencies simply do not fly through the Aether and impose themselves on low impedence inputs - and for audio, 10k is too low for capacitive coupling to enable enough signal to be coupled that it could be heard unless wired to a fully open Pink Floyd PA rig!