For a cheap easily replaced GFCI, which is a critical safety device, I
don't see why you would not just replace it. If it still trips when
presented with a test fault via the test button, it would appear to be
OK. On the other hand, clearly part of the GFCI circuitry is on the
fritz, so how lucky do you feel today?

Do I feel lucky?.... Sounds like a bad disaster movie... and those
never end good! I bought a tester from Home Depot today and it DOES
confirm that is not functioning properly. I may not be hooked up
properly, but if I'm going to undo the wires, I may as well put a new
one in.... I'm sure the new one will have instructions to insure I
reconnect the wires properly. Thanks for your post!