I'm back. And here's what I found.
The printing on this one is a bit different from yours, but I am hopeful that the mechanicals are the same. The clock does work, as shown in the first two photos. although I could have timed them better. it seems I'm almost exactly 30 seconds apart. the clock "ticks" rather than moving smoothly. I guess I wasn't sure what to expect. the cluster in my own Town Car is digital..
the 3rd photo is the back of the cluster. From the looks of things, it could be dissembled further, and the individual clock motor could be removed. If your problem proves to be the clock itself, and your instrumentation is built like this, I'm thinking this one may be an option.
Oh. and no fair making fun of my kitchen. that pot lid is clean. I just have to put it away.
Edit: that black cable extending from the bottom of the cluster is what controls the gear indicator. It attaches to the steering column with that white plastic clamp. *warning* the clamp is extremely brittle. this particular one is broken, as is the one installed on my car. Be very careful. I believe I have mine secured with bailing wire. (for real.)