How can I be sure my water softener is working?
Depending how they work, I'd have to wonder if the test strips are being thrown off by the "soft" salt ions replacing the "hard" calcium ones. But I really don't know for certain how those tests work.

You might try your "functional" tests on water that is drawn before the softener to see if the water after the softener behaves "essentially the same" or "significantly differently" in your testing: which would be one indication of "appears to do nothing" or "appears to work."