A dirty stylus can sound a lot like you described. Before you spend significant money on a fix which might be only guesswork, try properly cleaning your stylus.
The key word in the previous sentence was, "properly". I've saved a dozen "worn out" or "damaged" cartridges (including very expensive LOMC's that cost more than your whole rig) for owners who swore their styli were clean before sending them in for service/replacement.
They were wrong but happy to be proved so, since I saved them thousand$ in unnecessary replacement costs. Here's how:
Effective stylus cleaning. Search for "magic eraser" on this forum or Vinyl Asylum for hundreds of user testimonials.
Doug
P.S. Regarding a test LP, you've already used several. The records that your stylus isn't playing/tracking cleanly demonstrate clearly that there's a problem. No test LP will tell you any more. It's excellent that you're using your ears and real music to identify and try and diagnose problems. I own several test LP's and can't remember the last time I actually used one, despite setting up dozens (hundreds?) of cartridges. All my setups, tweaks and diagnoses are done using real music.
P.P.S. Yes, if your stylus is mistracking you may indeed be damaging your vinyl. Such damage is permanent and cannot be repaired. I'd advise against playing any important or valuable LP's before you find and fix the problem.