Villain Falls for or is Obsessed with the FL

This is probably the drama you're looking for.

Technically the FL is sharing another girl's body, and the villain is in love with that girl. 

The villain that falls for / is obsessed with the FL isn't a huge part of the story.

Obsessed in a non-romantic way.

There are several villains. One is more of an anti-hero and he's literally born out of another man's lust for the FL.

The ML is slightly villain-y.

The ML is a bit villain-y.

The villain is the FL's ex and the ML is a bit villain-y at times too.

The villain is low key romantically interested in the FL but also doesn't realize she's the girl he's obsesses with finding and killing because she saw him after he killed her parents (iirc?). 

An antagonist has a twisted mix of feelings for the FL: a bit of remaining affection for a childhood friend he once protected, and perhaps some romantic feelings, buried under hate and a desire for revenge.

iirc

The ML acts pretty villain-y at times.

iirc

Noble villain and morally grey SML.