OK, I am going to take a wild stab at this as there are only 2 times I have seen something like your conditions:

1) Ground issue, which you have ruled out

2) The emergency flasher knob is pushed in and your emergency flasher relay is missing.

There are 2 separate flashers needed for a '67 Firebird. One to make the turn signal indicators blink when the turn signal stalk is depressed either up (right side) or down (left side).

There is a 2nd flasher needed to make both turn signal indicators blink at the same time when the emergency flasher button is pushed in. I did not know this and I got always on turn signal indicators (both sides) when I tested the emergency flasher button.