Flashers 67 barracuda
I think your TS switch is all broken up inside. The older switches were complicated enough, but Ma added the 4x flasher switch into the turn signal switch, so now it has even MORE contacts inside.
The sure fire deal is to bypass the 4x flasher, just insert a length of wire into the flasher socket. With the turn signal switch centered, and the 4x flasher on, all lights should come on. If the rears do not, but yet they do work in turn or brake, then it HAS to be in the column switch.