I see no real problem with it.
There is precedent for a longsword-type weapon having both the Finesse and Versatile properties (Sun Blade). Beyond that, this is a +X Weapon that hurts you in order to use its properties, making it inferior in that regard to a normal +X weapon.
As a character levels up, taking 1-3 damage becomes an increasingly non-issue. One minor exploit is that if you attain Resistance to whatever damage type the sword does to its wielder (I'd go for Necrotic, since it is drinking blood), then you could use the sword in +1 mode at all times with no drawback. But the fact that it hurts you at all already makes it inferior to a basic +X Weapon.
If you want to shut down the 'Resistance' case, then make sure the damage dealt by the weapon is explicitly stated to be irresistible.
I would not suggest making it state that actual the damage taken by the wielder translates to the +X applied. While this would prevent resistance abuse, it would open the weapon up to Vulnerability abuse. If a character was made vulnerable to Necrotic damage and it followed that rule, they could boost the weapon to a +6.
This weapon would be fairly hard to abuse, because you must be attuned to a magic item to use its properties, so the blood drinking property could not be used on enemies. As mentioned above, it's basically inferior to a normal +X sword...though the Finesse on a Longsword does give Dex-based classes a viable two-hander option.
A few things other I will note in your description, however.
- You do not specify the damage type dealt to the player. All damage in 5E is typed in some way...I recommend Necrotic to represent it drinking your blood
- The description stating that it 'bind[s] to the wielder's wrist.' A player will possibly try to make the argument that a sword bound to their wrist makes it immune to disarming. This isn't a huge deal...but your description leaves that open to argument.
I will note that some degree of care should be taken if you hand this sword out at low level. I would rank it as at least 'Rare' and treat it accordingly (inferior to a +3 Weapon, which is Very Rare). At low level, the ability to spike your chance to hit by 15% for only 3 damage is well worth it for some classes--particularly bursty ones like Paladins who, at level five (with a single round to prepare) can drop 2 Attacks, 2 Spell Smites, and 2 Divine Smites in a single round...bumping their chance to hit by 15% would be a significant boon to such a character attempting to Overnuke a target like this. I am not saying this to note that it is unbalanced...simply that, like most high +X weapons, passing it out at low level can have a significant impact.
It has been brought up that using a Versatile/Finesse Weapon may allow some cheese with a Rogue's Sneak Attack by taking a 1 level dip into Fighter for Great Weapon Fighting. This is a non-issue because GWF only applies to the actual, raw weapon damage...not to bonus sources like Sneak Attack, Smite, etc. See the Question here as well as the following Sage Advice quote:
Q: Great Weapon Fighting and Battlemaster' Sweeping Attack: may I reroll the 1 and 2 of the superiority die?
A: Great Weapon Fighting works only with the damage dice of the qualifying weapon.
This pretty strongly states that it wouldn't apply to Sneak Attack either.