I remember going crazy with my 71 Charger with headlight closures as well.

Since you are able to close them manually probably not the plastic gear, but be careful, they break VERY easy ( ask me how I know..)

I've had the motor apart as well, check the contacts make sure they are not corroded and also making proper contact.

Also make sure they are adjusted properly.
If I remember correctly I had to adjust them manually to the point the doors would just start to close. That way there was enough travel when the switch was turned off.

If it turns out to be the relay, I think I have an extra lying around.

Keep us posted.
Good luck.