Symptoms suggest a failed bladder or tank. One mode of failure is rusting out on the air side of the bladder; usually near the water connection.
Check the air pressure in the tank with a tire gauge. If there is water on the water side of the bladder the air pressure should be same as the water pressure gauge. If it isn't you may have a bad gauge (which one?).
Consider replacing the water pressure gauge if it doesn't go to zero.
Pressure switches are much more reliable than pressure gauges or bladders.
Turn off power to the pump. Run all of the water out of the tank so water pressure is zero. Check the gauge. Does it read zero?
Rock the tank back and forth to verify that it is empty of water. If it has water then the bladder is almost certainly failed. Replace the tank if the bladder has failed.
Check the air pressure in the tank. The pressure should be the precharge pressure, which should be set at 2 psi less than the start pressure of the switch.