Rear tail light problem on my 74 Eldorado

So during the day over the weekend someone told me I had a tail light out. I checked it and sure enough it was out. The culprit bulb crumbled as I took it out so I replaced it. Now everything is working when the headlights are off, brake lights, turn signal etc, but when the headlights are ON, the tailights work, the turn signal works, but as soon as the brakes are pushed both right lights turn off. This is only happening with the headlights on and only on the right tail lights. Left is working fine.

When I first looked I saw this wire was dangling and I thought that might be the problem (though I don't even know what it does!), but even after I plugged this wire back-in I get the same result. Does anyone have any ideas?

Thanks!

Here is a birds eye view and then a shot of the bulb area where the loose black cable plugs into