Troubles with Turn Signals 67 Barracuda

You may have some wiring melted in harness and causing strange "cross connects." This is hard to run down, without just pulling the harness out for access and unwrapping it and inspecting it.

You have an evidence of "meltage?" Smell? Appearance, IE plastic damaged on the harness wrap, etc?

There is absolutely NOTHING that should cause the flasher to react. It comes (sourced) from the fuse panel, on the key "on" buss, through the flasher, and goes TO the column turn switch. With the switch centered, it should be "open." IT is possible you have something crazy in the switch.