63 dart question

Remove the dashboard faceplate (you'll have to remove the headlamp and windshield wiper switches; see here). Then you'll have access to the pushbutton mechanism, which should be lubricated every 55 years or so, whether it needs it or not. Use a good brand of spray silicone lube—quality matters a lot here; if you get the cheap stuff it can attack the plastic your pushbuttons are made of, and that's a bad scene. I use this kind. Spray the metal slides (behind the buttons) where they go through the rear wall of the button box. Put the straw in the park lever slot and spray to reach the hinge and spring. Operate the lever and buttons (you'll find them much smoother and more "clicky") and there's pretty good odds you'll find your reverse button stays pushed since its detent is now free to move into position.

Replace the light bulb while you're in there. The trick is to get some clear soft vinyl hose at the hardware store, 1/2" inner diameter. You need about 6 inches of it. Boil some water and bring it out to the car with you. Dip 1 inch of one end of the hose in the boiling-hot water for 30 seconds, then pull it out and immediately push the hot end over the bulb hidden amidst the pushbuttons. Once the hose slides over the widest part of the bulb glass, don't keep pushing. Let the hose cool for two minutes, then push inward and twist counterclockwise. Bulb comes out. Spray some of your silicone in the bulb socket. Install a new bulb the same way.