I do this for a living and have been for over 45 years. You have a galvanized tank with an air making system. Either your schrader valve is plugged up or your bleeder which is in your well in the drop pipe is plugged. The air control you have lets air out. It doesn't put air in. The first thing to try is replacing the schrader valve or cleaning the old one so it can allow air into the pipe once the pump shuts off. Once you clean or replace it, put a drop of water on top of the schrader valve and cycle the pump. When it shuts off, the drop of water should disappear as the schrader draws air. If it doesn't you will either have to pull the drop pipe up and clean the bleeder or change to a good bladder tank. Never put a cap on the schrader unless you are going to a bladder tank.