'67 Dart - No stop lights, turn signals on but don't flash

My turn signals stopped flashing, left and right lamps come on when selected but are on solid
The turn signals will stay on when one of the bulbs is out. Its not the only cause but one cause.
The turn signal flasher unit is a bimetal switch. When there is enough current through it, the bimetal gets hot and bends. This opens the connection and the lights go off.

I noticed on the dash D-C ammeter that the needle dips slightly for the turn signal current, left or right, but not when the brake pedal pressed;
The ammeter indicates current flowing into or out of the battery. In other words Battery Charging or Discharging.
checked and no stop lights.
The brake lights are out, so there is no current flowing out of the battery.

Turn signals should only work with Key in Run.
Brake lights should work all the time.
Parking lights (front) work with the headlight switch in Park only.
Tail Parking Lights stay are on with headlight switch in Park and Headlights on.

Double check this, but I beleive your car has the 1034 double filament bulb for turn, parking, brake.
They ground through the side of the socket.
Since the bulbs light up for turn signal, the ground connection is working.

After checking all the bulbs and the fuse we can check wiring.
I agree with Mike. Turning on the hazzard switch is a good check. Hazzard uses a different flasher and wiring route, but the same filaments in the bulbs.

How Turn Signals Work
67 Barracuda Turn Signal Dielemma