To answer your question about distance, I dont have a “drill”, if you will, but my main objective for having students get more distance with the driver is to make sure their grip pressure is as light as possible, thereby reducing any stress and muscle tightness in the hands and forearms. If they can keep a very light grip on the club, they will increase their swing speed just by doing this.
I also have them try to feel that the right arm is straight on the follow through swing, thereby increasing their extension through the swing. I find that if I can get them to do that, their extension on the backswing will also increase, giving them a larger arc and more club head speed.