Well, I'm going to go into, probably way too much depth for this; just a warning.
I looked into the sword that you were talking about, the mission retrieving it, it's original wielder and it's effects. In Oblivion, it's enchantment/effects drain fatigue and deal fire damage. It's also an Elven Claymore.
Now, in Skyrim there is no Fatigue, so there needs to be something that makes up for that. As Fatigue essentially effects the effectiveness of the user's actions, it'll be rather difficult to really capture that. Now, when I think of Sinweaver, it's name to me describes it as a weapon that fills it's target or wielder with sin, darkness, evil etc. I think this because of the actual name when you dissect it. Sin is relatively easy to understand - the act of doing something evil, or something considered a transgression. Weaving is just as simple - combining or interlacing the fibers of something to create something new, something stronger. So essentially what I am thinking is that the weapon fills whosoever uses the weapon, or is targeted by it with some sort of power while also destroying them as a side effect.
So, for Sinweaver what I am thinking is that it's first enchantment would be Absorb Stamina. This substitutes Drain Fatigue as best you can in Skyrim, and also fills the wielder's stamina reserves. So right there, you are weaving in the energy of another into your own - and in my mind, this could be seen as a sinful act. Stealing the very energy of another person? Seems pretty unnatural and rather dark if you ask me. This leads to the next bit, what negative effect would the sin have on the wielder for doing such a thing?
While in Oblivion it's secondary enchantment is Fire Damage, I don't necessarily see that as a negative effect on the user. So I am thinking the Fear enchantment instead. I know for certain, in my experience anyway, when you are wielding a two handed weapon and your foe turns tail and runs away, this can be quite the nuisance. The last thing somebody with a heavy weapon and even heavy armor wants to do in the midst of battle is attempt to run down their opponent, only to lose all the stamina they just absorbed in trying to catch them. Nothing short of annoying.
As an alternate option, filling the opponent with sin in order to negatively effect them, is rather difficult. So I am thinking the first enchantment could be the Fear enchantment. Why? Well let's think about it from a role-play perspective. Almost all enemies usually stay in combat until they die or you do, but what if something supernatural showed them what their future holds if they remained in combat with you? Something that says "hey, you should get the hell out of here before you die." What's more dishonorable than starting a battle only to run away from it? I would say that's a transgression. So what's price of such a warning, such an "edge" in battle? How about Fiery Soul Trap? Not only does it burn the foe (like in Oblivion) but it also damns their soul into the prison of a soul gem - only to await their inevitable doom once their soul is destroyed to be used for charging or enchanting.
Of course this weapon must be an Elven Greatsword, thus being an artifact of ES: IV
(This was a rather challenging thing to construct in my eyes.)
Now for Sineater, it's twin so to speak.
To me this sounds like a weapon that absorbs and uses the sins of the target to destroy them, or perhaps harnesses the power of the sins of the wielder.
So, focusing on affecting the target, I'd say the first enchantment needs to be Absorb Health. While I was thinking about Absorb Stamina, I feel like the sins of a person would be more connected with their life force rather than their energy - and if Sinweaver has Absorb Stamina on it already, it wouldn't make sense for it's twin to have the same. So, Sineater eats the sins of it's target, which in turn, drains the life force all the sins are attached to.
As for the second enchantment, the enchantment that uses the target's sins against them, I see one of 2 options: the Paralyze or Chaos enchantments. For Paralysis, what I imagine is the target's sins enveloping it's original host, freezing and overwhelming them entirely - essentially taking a physical toll on them after being removed and released from within. For Chaos, I see this as a bit more Roleplay friendly. Because it has a 50% chance of using Fire, Frost and Shock damage, I can see it as somewhat measuring the sins of your opponent. For example, you use it on one enemy, and it only does fire damage; then said enemy has fewer sins than another. Then you use it on a different enemy, and it does Frost, Fire and Shock; so this enemy is an incredibly sinful bastard.
Now to focus on the alternate option, affecting the user. For me, this requires a little bit of work, however it's worth it for the role-play and the immersion. Let's say that this weapon, in order to be used to it's full effect, the user must take something in order to link their soul with that of the very weapon. A potion of Fortify Two-handed and Damage Health; this potion can be made with Troll Fat, Dragon's Tongue, and Deathbell combined. This is to fit into the roleplaying of the weapon's effects. Absorb health, in order for this weapon to work, has to harness the sins of it's user - which are attached to their life-force. So, popping this potion before use of Sineater would act like you are removing the sins from your life force while linking with the weapon, in order for it's power to be harnessed.
Now, once your sins are connected with the weapon, you'll not only get a boost for your two-handed ability, you'll get access to the weapon's enchantments. The first enchantment could be the Paralyze enchantment. Like the alternate version of this, the user's sins will envelop and immobilize the target, physically affecting them. It's second enchantment could be, again, the Chaos enchantment. Just like the alternate, this could be a measurement of your specific character's sins for that use of the weapon. The second enchantment could also be any of the Elemental Damage enchantments. I'd say if this is the option you go with, I'd pick whatever element you think aligns with the character that you are using at the moment. Perhaps a Battlemage would be using Shock, as Shock does damage to both health and magicka; which could be viewed as a connection with the users magical prowess. Maybe if your character is a Nord, it would be Frost, to represent their frost-born skin and souls. The options go on.
As for the type of weapon that this would be, I'm thinking something that could be looked at as the opposite of Sinweaver - an Elven blade. Something just as elegant, but somewhat of a yin to Sinweaver's yang. I'm thinking an Ebony Greatsword would be particularly perfect for this. It's pretty, stunning, and dark - opposite of Elven crafts' light.
Sorry for the wall of text
I hope you like the ideas!