Take a look at Dual service manual for 1218, found here: http://www.vinylengine.com/library/dual/1218.shtml
On Page 2 there is a wiring diagram for the cartridge wires. You will see that there is a muting switch that shorts the cartridge out when the arm is on the rest, or when the changer mechanism is operating to change records. It may be leaving itself shorted all the time on one channel. You will find muting switch under turntable, and you already appear to know where it is. Take a look at the little metal shorting arms and see if they split open when you put the tonearm on the record. If they don't do that just stick some tape between them. This looks to me as if the shorting switch might only be opening up on one side.
Before I started to cut stuff up, I would scope out the muting switch, and be sure that it is opening up when the turntable is playing records. If that does not take care of problem, you may have to start disconnecting wires to find where the short is. If you take a look at manual, this unit apparently does not have a separate ground wire. The right channel shield is connected to the metal turntable chassis and the chassis ground is carried to the amp on the right channel shield. You do have the option to use a 5th wire, just disconnect right shield from chassis and run a separate 20 gauge stranded wire to the the chassis, and connect that wire to chassis of amp.
If you can open up your RCA plugs and look inside of them, be sure that they are not internally shorted. See if the caps unscrew.
And good luck from that muted, but well shielded old guy,