Main thing to verify is that the particular footswitch and cable you have is specifically for that amp. If it is not the right one, that is your problem as there is very little compatibility among footswitches.
Looking at the schematic, plugging in JUST the cable shifts operation as you observed. This uses a crazy circuit in which AC is applied on the cable and the footswitch has a diode each direction that causes the amp to shift between the states. when you plug a bare cable in, it disconnects the internal diodes and switches for control at the amp, hence the signals float.
You should investigate the footswitch to make sure it is the right type with the diodes resistors and caps that is for the Fender Deluxe.
You can find the schematic for the amp here:
Look at the circuitry of the switches on the schematic at C8. Your pedal should have similar circuitry. Assuming your cable is good, the problem would be in the pedal itself.