I'll mention what I've done, not knowing for certain if you have something like that in your game, but perhaps you can use the idea in theory.
Instead of having "life drain" in that it removes skills or attributes, I have done where it "adds Fatigue". I used the Fatigue aspect as a way to make it so people didn't just fight their way through everything all of the time, or sleep in their armor. Each point of Fatigue reduced from their die rolls for ALL skills and attributes (except resistance rolls to limit damage). So 1 Fatigue is a mild annoyance. They just minus 1 from the total each time they make a skill or attribute check. But once you start adding up the Fatigue to 2, 3, 4 or more, they start REALLY noticing the effect when it's on every die roll.
So with that said, I would have an undead like a Wraith inflict something like 3 Fatigue each time they successfully hit a living being. Or if 3 is too much, make it an even 2.
Why not make it based on quality of hit? Because of the effect of Fatigue and the slow rate of recovery from it. (Need extra sleep above 8 hours for 2 consecutive sleep cycles to recover 1 Fatigue. Each consecutive day after the 2nd with extra sleep will recover 1 more Fatigue. If a person doesn't get more than 8 hours sleep on consecutive sleep cycles, the requirement of 2 consecutive sleep cycles starts over.
It won't take too long before a character's Fatigue makes them so completely worthless that they flee in terror. "What do you mean I have to minus 12 from my rolls?"
But you can also vary the effect of certain undead. Something like a zombie or ghoul could just do claw/bite attacks, but with the added effect of the character needing a Stamina roll of Moderate (for a scratch) or Very Difficult (for a bite) to avoid being infected with a disease that provides an ever-increasing -1D to their Strength/Constitution (whatever you might use) every 30 minutes or 1 hour, and when the penalty exceeds their attribute, they pass out and may go into a coma until healed or dead.
Or you could have, as you mentioned, a "cold touch of death" effect on the target, in addition to the damage inflicted in the attack. The "cold touch" pierces to their soul, so to speak, and counts as one point of "death" upon the character. When the points of "death"...an ever-increasing coldness in their body, even if it is sunny and warm out... reach 3 they lose 1D from EACH attribute. When they reach 6, they another 1D, and so on every 3 points of "death". When an attribute like Strength/Constitution or Intelligence reaches 0 or lower, the character dies. They can eliminate the "cold touch of death" on them by defeating the undead that inflicted it upon them, or by going to a holy site and resting there under the care of a priest. Each 6 hours of rest and care on a holy site will remove 1 "death" touch.
And certain things could provide protection against these attacks. Perhaps a holy relic/charm may either increase their resistance against undead attacks, making it harder for the undead to actually inflict an attack that could spread disease/increase fatigue/deal the cold touch of death. Or perhaps a blessed item could cause additional harm to an undead, or maybe it could resist a certain amount of evil taint...say by negating the first 1 or 2 or 3 instances of the undead's special attack.
Another thing you could do is make it so that things like Ghosts could attack by choking or squeezing a person, ignoring the character's armor and ability to "dodge", since it is a relatively unseen attack. Of course, if the character's could do something to make the ghost visible (smoke, dust, whatever) they could strike at it, but with an added difficulty to hit (say +3D for a mostly incorporeal ghost, or +1D for a mostly visible...without assistance...ghost that can be seen relatively easily by is still a less than corporeal being).
I would suggest deciding upon each type of undead creature to see if certain things are more beneficial. Perhaps silver works particularly well on ghouls, but is just like any other weapon against zombies. And vampires might be more harmed by piercing WOOD weapons rather than anything made of steel. Perhaps steel only does 1/4 damage or 1/2 damage to a vampire, but a wood piercing weapon does triple damage. All other weapons would do normal damage. But you could also add enhanced regeneration for damage done by non-specialized attacks. I always use an old movie I saw called "Monster Squad" as a basis for this. They blew Wolfman up with dynamite. Litterally blew him to tiny pieces and bloody chunks. But the pieces magically merged back together over time and the wolfman came back to life after a while, because the attack wasn't done with silver. But the silver bullet later completely killed the wolfman when he was shot in the chest. The damage was enough it would kill a person anyway, but this time, wolfman stayed dead because it was with the item that made the damage "real" and not able to be magically resisted. Ghosts could be seriously injured by something as simple as holy water tosses in their direction. The tiny water particles would be like acid to a ghost. Or maybe smoke from a burning censer would cause ghosts to be kept away, and stop certain undead from coming into it (like a wraith) but mean absolutely nothing to a vampire or ghoul.
Hopefully this gives you some ideas.