Troubles with Turn Signals 67 Barracuda

First, assuming your system is fairly original, IE original style flasher(s) and no LED replacement lamps, there are TWO flashers. One for hazard (4 way) and another for turn

BOTH flashers MUST have lamps hooked to them to flash. Usually, on older flashers AND depending on how "heavy duty" (part number) the flasher is, a burned out lamp will cause the unit to either flash not at all or change speed dramatically. So check lamps

Visualize your TS switch. It routes several sources and loads several different ways

What goes into the switch:

1..Hazard flasher power output, hot at all times (EDIT INCORRECT before 70). On the earlier cars like yours, hazard flasher feeds the separate dash mounted hazard switch
2...Turn flasher power output, hot only with key "on."
3....Switched brake lamp power coming from the brake pedal operated switch, and operable at all times

What comes out:

1...Left rear stop/ turn lamp filament
2...Right rear stop/ turn lamp filament
3....Left front turn filament,also branched off to feed instrument panel indicators and or fender indicators
4....Right front, same as left for indicators

WITH the TS switch in center, the switch routs power from the brake light switch to BOTH rear lamps

One thing you can do is to remove both flashers and temporarily connect a jumper wire across the two flasher terminals. That way, when you activate the turn switch or hazard switch, you can walk around an observe that the proper lamps are lighting, and bright.

OTHER tricks: Lamp sockets especially rear, that become ungrounded can be tricky, because with no ground, the two lamp filaments (tail and stop/ turn) are now in series, and will light 'sort of' improperly.