The headlight system on the '67 cougar is vacuum operated. Most likely you have a vacuum leak somewhere in the system or possibly a bad headlight motor. If you have a hand-held vacuum pump, I would pull the vacuum lines off the headlight motors and use the hand-held pump to check those first. Then I would start checking for vacuum leaks. Don't forget about the vacuum canister under the driver's side fender as well. Sometimes they develop rust holes and won't hold a vacuum.
There is a little headlight vacuum modulator on the driver's side fender apron that might be culprit as well. But I would check for vacuum leaks first.
Have they ever closed? There's a plastic push-in switch on the vaccum junction/control block in the engine compartment, near the front on the drivers side, if memory serves me correctly. If that switch is pushed in, then the headlight doors will stay open all the time. I think the most common problem with the headlight doors is that they won't open if there's a vaccum related problem, hence the switch that will hold 'em open. I'd check that push-in switch first, I'm pretty sure it has to be in the 'out' position or it will hold the doors open. It's been almost 9 years since I had my 67 Cougar, though, so I may be wrong. . .