I had a problem with my US Turbo Duo (all original, bought in '93 as TTi went under) where the yellow composite RCA plug was faulty, so the video would fade in & out and I had to move it around all the time so that it would stabilize, etc. I got tired of this so eventually, I decided to replace it. I cut it off, got a RCA plug from some other cable, did the needed wire stripping, twisting of the copper wiring together and taping, etc. A couple of years later though, it started giving me problems again because I had been lazy and didn't solder the connections together the first time around. Twisting copper wires together isn't good enough and with gradual movement, conductivity becomes an issue, etc. Point is, if that is the problem and you do this, make sure to solder the wiring so you won't have to deal with that issue again (and yes, I finally did it in my case, and I cleaned the contact tips with a SOS pad - no more issues now). The other possibility has been pointed out about having to open the unit up and resoldering the AV Out mount.
EDIT: Oh OK, I skimmed over the first time but now see that what you pointed out definitely indicates the problem is at the AV out area. A contact issue when the AV plug is fully inserted. Well, if you can try to clean the pins with a trimmed q-tip or whatever and alcohol, that might be helpful (was in my case because of dust accumulation given that I leave most of my unused gaming hardware in the attic), but it sounds more like the tightness of the receiving pins within the unit are worn and don't make good contact after the cable is fully inserted.