I'm not sure I understand. YOu have a storage tank that you expect to hold air and water with no bladder separating them? Usually, the storage tank has a bladder to keep the system pressurized while the pump is off. Any tank that has air and water in it will eventually disolve all of the air into the water, and you'll only have a tank full of water. If the water doesn't "disappear", it would typically bubble out when you run the water through the opened faucet.
Does this tank have an air valve on it? If it does and you open it while it exhibits the problem you indicate, does water come out of it? If so, then you need a new tank.
With the pump off, a valve opened in the house to relieve the pressure, the tank should measure a couple of pounds less than the turn on setting of your pump. You should be able to use a (quality - hopefully accurate) tire pressure guage to check this.