Test 1: Turn off the pump. Open a faucet, so that the water pressure is zero. Normally at that point the tank will be empty of water. It will be relatively light.
Test 2. Assuming your pressure switch is 30/50 psi. Turn off the pump. Run water, so that the water pressure drops to near zero. Leave a faucet open. Measure the air pressure. Pump up the air pressure to 28 psi. (If you cannot, tank is bad, or your compressor is bad. ). Put the tank back into service for a day or a week or a month. Repeat the test. The tank should not have lost air in that time.
Put the tank back into service.
Note that for adding air, if you buy something, you want it to have a CFM or SCFM rating. If it only has a PSI rating, it may take a long time to pump up a car tire -- let alone a big tank.
The symptoms alone make it unlikely the tank is good, but I understand wanting to confirm.