SDM0, I was just explaining this to a friend yesterday. Funny.

Unfortunately, they do get rather hot. They're the only coil in all EM pinball machines that gets locked on whenever the game is fully powered up. It is a necessity to the game's function and it will kill off the entire game without it, unfortunately. It was not a very good design but it is what it is today. The best thing you can do is replace this coil before it even dies (it will help what you are describing and definitely be safer), and if you keep up with the frequent powered-on-ness, I always recommend that in a commercial setting that you replace the coil every few years if it shows signs of needing it.

The coil you are looking to replace is a Z-28-1150, and I recommend replacing it with the better option given, which is a Z-28-1200 - which will run cooler. These specific coils are already built (and were back then too, of course) to run for these long periods of time, otherwise they'd burn up in a matter of minutes. Less power to move a tiny relay = less heat. Anyways, these replacements have some extra windings on them which gives it a marginally even-LESS amount of power, which in turn, again, makes it run even cooler.

Check it out:

http://www.marcospecialties.com/pinball-parts/Z-28-1200

Best wishes and good luck with your Williams machine,
Otaku