1969 dart turn signal switch issue

Sounds like it is indeed something in the TS switch.

Just for info here is how the hazard switch works in those

With the TS switch centered, brake switch power is routed to both rear lamps. The hazard switch has 4 terminals, and when off none are interconnected. When on, all 4 are connected together

1 terminal receives power from the 4 way hazard flasher
1 wire leads to the brake light switch power going into the TS switch
1 wire leads to the TS wire feeding front left lamp
1 wire leads to the TS wire feeding front right lamp

So with hazard switch engaged, the flasher feeds power to those three points. For it all to work, you foot must be off the brake, and the TS switch must be centered.