So: what the fuck is the Dying Man?
Well, from what I've been able to gather, he's a body-jumping parasite. Have you ever seen that movie Fallen, where Denzel Washington has to fight the demon Azazel who can transfer bodies with just a touch? He's kind of like that only it sort of takes more than touch to transfer. It takes dying. Hence the name.
Imagine a knife-wielding maniac attacks you and you, in self-defense, kill them. If they were possessed by the Dying Man, well, you're screwed because now you're possessed by the Dying Man. I have no idea if there is a way to get rid of a Dying Man infection, because all reports of infections I've heard end in the possessed dying. The best thing you could probably do is go somewhere completely isolated to kill yourself, so that the infection isn't passed on.
Now, here's where it gets complicated. You see, you might not be able to go anywhere; you might be completely controlled by the Dying Man. Or it might just mess with your head. Or it might unbalance you. Or it might lay dormant.
Because there isn't just one Dying Man, oh no. There are many. Apparently, it used to be one big thing and then it was split into many smaller things. Because that always works out great.
In any case, being infected by a Dying Man usually means your body will either immediately be taken over or slowly be taken over. This also applies to the rate your body degrades - oh yeah, I forgot to mention that the infection also comes with degradation of your body, mutation, decay, and, eventually, death. Because the Dying Man is always dying.
Another thing to consider is that apparently all the different Dying Man have different names and personalities or something. I'm really not sure about that one - all of the operatives I've talked to who have come in contact with the Dying Man have avoided talking with him, probably for fear of becoming infected themselves.
The Dying Man is one of Them that you should not, I repeat NOT kill. Killing it in a populated area will only release it to be passed on. Probably to the person who killed it.
Your best bet if you actually capture someone infected with the Dying Man is to take them to an isolated area where you are sure there is no people for miles around. Then a slow acting poison might work best and allow you to leave and get far enough away to avoid infection. In any case, it's a lot more complicated than just shooting them.