Electrical issues.

vpmopar said
Turn the key on and the turn signals work perfectly, right and left front and rear and the indicator lights on the dash work correctly as well. with the key on, turn on the left or right signal and it works fine until you turn on the headlights.

Lets follow the rest of the path here. This is based on '67 with the hazzard switch removed.

Using the left turn signal as an example. Current flows to the flasher, then to the column connector and up to the turn signal switch.
The switch connects the rear front turn signal lamp seperately. This is because stepping on the brake will overide the rear turn signal.

vpmopar said
with the headlights on, all the exterior lights work correctly but both turn signal indicators in the dash will flash and the high beam indicator also will flash.

Here's where the problem is. So lets break it up.
This does not happen with the parking lights on alone?
If so, then its related to the headlight circuit.
If it happens with parking lights on alone, then its parking/tail light circuit related.

The fact that both indicators are blinking means power isn't flowing through the turn switch but is finding a short cut.
Also, to get the indicators to blink, power must be flowing through the flasher.
Since the high beam indicator is blinking too, even on low beam, something must be connecting it to the turn signals.

vpmopar said
If you hit the dimmer switch, and turn on the left signal the right indicator light on the will flash, and if you put the right signal on. the left indicator will flash, but all the outside light work correctly.

Do the turn signals respond at all with the headlights on low beam?

FWIW. On '67 Barracuda, the high beam indicator wire comes from the bulkhead high beam terminal.