At the risk of adding more confusion, rather than resolution here, I will interject the following thoughts. These thoughts are relative to static driven instruments, not air data, electronic, and repeater types:
An altimeter being "sensative" or not, is a reference to it's resolution of indicaton of altitude. I would be amazed to hear that anyone posting here has flown a certified aircraft with a non sensative altimeter. They've been "gone" for decades, as far as I know. If you have a non sensative altimeter, one or two characteristics will be apparent; the finest resolution is 100 feet, rather than 20 feet, and the indication of 1000 feet of altitude change may be other than one whole turn of the "hundreds" pointer.
As for the encoding altimeters, though I'm sure there are many "new" designs, the encoding altimeters (as opposed to blind encoders) we will likely find in our light aircraft generally have a glass disc which looks like a tiny CD at the back of the instrument. This disc is read similarly to how a CD is read, and that is where the alitiude information comes from. If anything, this disc would damp out errant pointer motion.
I would expect (based upon my knowledge of North American requirements) that aircraft equipped with encoding altimeters, and mode C transponders must have annual or biannual altimeter calibration. Certainly a flaw in the operation of the altimeter itself would become apparent during such recertification. A non sensative altimeter would not be approvable in an aircraft equipped with mode C.
If you have a 500' difference in indicated altitude between two altimeters in one aircraft, at least one of them is not airworthy.
As for the original question, the altimeter in question needs a visit to the instrument shop. Ther are internal opposing forces designed into the instrument (steam gauge types) to cancel out such indications. A part of one of these mechanizms is not working as intended. The altimeter may still read accuractely when not wandering, but requires fixing anyway. There is nothing to be done outside the instrument, as long as you are sure that there is no static blockage.
There was a reference to applying air pressure to the aircraft's static (or pitot) plumbing. NEVER! unless ALL pitot static instruments are completely disconnected. It is not possible to regulate a compressed air source finely enough to not damage an instrument. Recall that half of our 14.7 PSI atmosphere is below 18,500 feet. That means that at 18,500 feet, the pressure is about 7.35 PSI. That's 18 1/2 turns of your hundreds pointer. so about half a PSI is needed to make it go 'round once. Can you regulate the pressure source to 1/2 PSI? Can you even measure it? Use care with instrument handling....