Its a very common problem, but I think its a dealer only part. I know several TR vendors sell them, and I think Red Regal T on this board sells them too, but I could be wrong. I doubt its connected to the alternator, it would never shut off, ever. One thing you can do is unplug the relay and remove it and shake it. If it sounds like its full of rocks, its bad.
Mine rarely let the fan run when I shut the car off, even after I replaced it. The car has to be at a certain temp. for it to happen, and with my aluminum radiator, 160 stat, and dual fans, its never been that hot.