Is this only with the Right Turn Signal or does the Left Turn Signal have the same effect? It would only stand to reason that if the Headlights flash with the Right Turn Signal that they will flash with the Emergency Flashers.
The Front Turn Signal/Run Lamp is supplied Ground through the base of the Lamp that bolts to the Valance, but if Ground were an issue, it would travel through the Run Lamp Circuit and find Ground, which would cause the Run Lamps to flash and not the Headlights
The Headlight Circuit is isolated from the Run Lamp and Turn Signal Circuits, with that being said do you have a DVOM? (Digital Ohm Volt Meter)
I would check the resistance between the Right Turn Signal Wire (Dk Blue) and the Headlight Wires (Tan -Lo Beam and Lt. Green – Hi Beam). Hint: if the High beam indicator isn’t flashing while the Right Turn Signal is on it is probably the Low Beam. If the resistance is low then the two circuits are somehow shorted together. They could be pinched, melted, pinned wrong at the cowl connector, etc. Inspecting the front headlight harness carefully might lead you to where the trouble spot might be. I do have Electrical Schematic information on the 69 Camaro if you need to know the correct pin assignment of the cowl connector.
I hope this helps and please feel free to contact me, I will do my best to help you.
Sparky