Disclaimer: This is all just my opinion. ;3 I'm not an expert in Dardunah Lore

Well think of it this way... If you allow this player to get away with tons of additional character points to spend by picking crippling drawbacks, they have to know that those WILL come back and bite them in the ass some time around.

Take my character for instance... he's a turtle with moist skin.
Fine I thought, free points! Nyeeee! I need to carry moisturizer with me all the time and use my free time to make it. And deserts are kind of a no-go.

So if this player decides he wants to be pitted against a mortal enemy that is by far stronger then him AND haunts him in his dreams, don't let him live with the feeling of safety that you wouldn't let him get killed off by it... Cause he just might.

But remember: Demons don't just jump out of bushes and attack people and they are generally huge dicks.
They like killing people, but what's cooler then dead people?
Utterly terrified people who have lost all their sense of what is real!

So I would suggest that instead of trying to kill him, the demon would perpetually try to interfere with what he's doing when he hopes that something interesting/amusing could happen.

A demon who has taken liking in torturing your player will creep around the corner, waiting for a moment to pull the perfect prank.

Make sure that you remind the player that that is the doing of the demon, but too late for him to avert anything.

For instance, the player lands in jail due to the demons doings. While sitting in the dark, damp cell suddendly there's a pair of red eyes in a dark corner of the cell, or just a pair of hellish claws reaching out of the darkness. The demon starts a short conversation with his victim, making fun of him and congratulating himself for a job well done, and then disappears.

Generally you could handle both types of influence these ways:

"Demon Plagued" Type Influence:
Unlike the normal demon plagued drawback which would only apply to one or multiple stats the character has like some sort of debuff, the demon that plagues him and is his enemy would be an intelligent entity really dead set on plaguing this player that tries to manipulate all sorts of actions.

He might wait for your player to fumble or just have a partial success on an action and then sometimes intensifies the negative outcome or just manipulates it maliciously.

Say your player attempts an action that the demon might want to sabotage. Let the player roll, and then subtract the number of successes from the Essence stat of the demon. The remaining Essence will be how many dice the demon has to sabotage this action.
If your player fumbles, the number of fumbles will add to the number of dice the demon has.
Depending on how many successes the demon has when rolling, the outcome for your player will get worse and worse.

Examples for Sabotage:
Having a player make a healing salve, but the demon fools him so he creates a potent toxin.

The player talks to some important person and tries to impress him/her with his speech skills. The demon could sabotage him to spout profanities instead without knowing of it.

The player does a healing ritual, but because the demon chooses to interfere and wins the challange instead of healing pranah, tainting demon energy flows from his hands, tainting their wounds.

The player tries to craft an item as a gift for a friend, or enchants it.
If the demon decides to interfere and wins the challenge, the object might be enchanted with a negative effect. For instance you make a beautiful necklace for a good friend you respect, but the demon enchants the necklace to make everyone wearing it slightly dislike you, not see you at all, see you as their mortal enemy, fall ill or for when the demon gets a spectacular success get changed into a creature (some sort of pet), another type of janah, gender or simply make them absolutely hideous.

"Enemy" Type Influence:

The demon might decide to take the form of objects, janah and creatures in the players surrounding, with the goal to fool him, harm or ridicule him.

Examples:

The player enters some sort of contest, the demon turns up as a contestant, sabotages the player and wins.

The player comes to visit a friend of his, of which he doesn't know that he has been slain/restrained/magicked away by the demon, who appears in his place and gives false information, cursed gifts and potential harm.

The player is lost in the desert and sees a Suthra in the distance that seems to drink out of a waterhole. When he reaches the waterhole the Suthra is gone and instead of a waterhole there's a desert spider nest full of angry spiders.

The player comes to deliver an artifact retrieved for a lord, when suddendly a dromi, or a small janah steals the item from him and places it in a hard to reach place, the hands of someone else or just carries it away (Sidequest).

To make things interesting and take a little heat off this poor player foolish enough to tempt the demons, the demon might want to avert especially bad things that happen to this character which he wasn't resposible for and don't especially entertain him, or help him out of situations that simply bore him ("I always thought it'd be thrilling to watch someone shrivel and die inside a dungeon, but this taking far too long for my likes..." and suddendly the player finds the keys to the cell doors in his pocket)

Or maybe another demon starts playing with his victim, so the original demon fights to defend his turf.

Btw: Don't worry about the demon trying to fight the player in whatever form he might take at the time, may it be an actual demon spawn, the image of a janah, a suthra or an animated object (spoon!). When the demon decides to fight his victim and is stronger than him, just make sure he deals more subdue damage than fatal, that way he just knocks him unconscious.

Just make sure there's an interesting reason for it:

Be it that the player decides to enter an arena fight and the demon mimics an arena fighter to show the player who's boss.

Or he tries to convert a local village to follow his cause, but the demon has taken the form of a shaman and took control over the village and persuades them to stay strong and defy the great oppressor.

Or maybe the player was invited to an important event, one that is maybe held to honor him/her, so when they take a drink to calm down... suddenly a flask on the table tries to execute a sneak attack and knock him/her unconscious. Then everyone finds them in their room, and thinks they went down from drinking too much. In effect much of the respect they had for them may diminish.

TL;DR for ArmaCoyote:
The whole point was that it would be boring to handle the demon plague as the standard debuff it is and then just having him pop out once in the while to kick your player in the nuts, but connect both drawbacks into one powerful storytelling tool that allows you to use the demon to create unexpected mischief whenever you want, and depending on how severe the drawback is.

Boy, you're certainly nailing not only the spirit behind the game, but the rules as well. Glad you're posting.

-Aaron

Thank you very much.
I wasn't sure if what I wrote up had any parallel at all to your idea of demons, but I'm glad they're not just mindless monsters, but also can be clever and cunning bastards (great opportunities for storytelling!).
I'm glad you like my posts, I'll try to edit my post to elaborate even more on it!