Always, or just when using water?
You almost surely need a new diaphragm pressure tank. You could do further testing, and for now work around for a while by adding air.
Your new pressure tank should let the pump run for a minimum of a minute, and that usually means a tank with a nominal size of 4x or more of the GPM from the pump. A 44 gallon tank is usually a good size.
With all water outlets shut off, the pump begins to pump. Of course, pressure switch contacts are pulled in, and gauge is showing steady rise in pressure. Once it hits cutoff pressure ~55 psi, the contacts in the pressure switch open (pull apart) and the pressure swings dramatically down, then back higher and then back down, (you get the idea). The pressure ultimately settles down, much lower than the original. Not long after the pressure switch engages again, causing the well to pump once more.
The drop is so quick (less than a second) that it can't be a leak above ground, plus, the well pumps up to pressure at a seeming normal rate. If something were leaking that largely, it seems like it would take much longer than normal to pump up.