Thanks for the advice everyone. I went ahead and removed the firing pin, but when I tried to dry fire the gun the trigger would not reset. I reinstalled the firing pin and it worked again. It looks like if the hammer travels too far forward that it won't reset.
OK, so it looks like I have a couple of options now. I can keep dry firing the gun and just watch for breakage of the firing pin. Since I don't use this gun for self-defense, breakage of the firing pin is probably not that big an issue.
Secondly, it looks like I can tighten down the firing pin set screw so that the firing pin won't move when struck by the hammer. Will immobilizing the firing pin protect it from breakage?