I'm gonna give these scores based off how much the show is focused on the internal struggles or personal growth of the character on a scale of 1-10
Flower of Evil - 5 - This is really more of a suspense/mystery but people love it and there is moral struggle. the positive of this one is that it is very deep and intense psychology going on with that ML that has everything to do with an inner struggle between good and evil.
Cheese in the Trap - 10 - The quintessential morally gray ML. Seriously, this is like the classic prototype for morally ambiguous. However, he's not battling in himself between good and evil, but rather more nuanced issues of 'right' and 'wrong'. He is a 'good' character. He's just a character that his exact level of 'goodness' is up for debate. The entire show revolves around this inner struggle and his challenge to overcome himself and grow.
Well-intended Love - 5 - (if you can handle the fury of the bad quality, the ML is pretty spicy.) The show is less focused on his moral struggles, and he is a pretty recalcitrant ML, but the struggles become real to him as he realizes the FL isn't liking his choices and behavior.
The Rose - 8 - (one of the awesomest most messed up shows I've ever watched, probably my all-time fave drama or at least tied for first) Several main characters go through moral struggles here and the show is mainly focused on how they have to grow to make it through, even though they are really resistant to change.
A Korean Odyssey - 6 - This is a less "heavy" show in the psychological department. But seeing as how the ML wants to literally eat the FL when they meet, I'd say he goes through a lot to get to the point where he's gaga over her and would do anything for her.
Til the End of the Moon - 10 - This entire show is about 'the battle within' between good and evil and right and wrong. It's not psychologically heavy-duty, but neither is it light by any means.
That Winter The Wind Blows - 7 - ML starts out trying to play a long con on a blind FL. Then he gets himself ensnarled in her life to the point where he has to make hard choices about how to proceed. He is so accustomed to being morally gray that his love for the FL pretty much shreds him internally.
Jealousy Incarnate - 6 - ML starts out as a real jackass and has to develop a whole other side to himself in order to cope with a new circumstance. He's not exactly 'morally gray' in the classic sense, though. However, the show takes his struggles seriously.
Jotei - 7 - Nearly every character in this show has some shade of moral ambiguity. The ML and FL are both morally gray and must learn to grow as people. However they didn't go as hard on the 'deep psychology' as they could have, and their character development is pretty predictable.
Love Between Fairy and Devil - 7 - The entire drama is centered around the moral ambiguity of the ML, however it was a cheesy wuxia that doesn't really cut right down to the psychological core of things. I did like it a lot though.
My Heart Twinkle Twinkle - 10 - Dear god "the chicken man", as I call this guy, was the most messed up morally mixed up ML I've ever seen, and he does NOT want to change, but he HAS to. Through the show, the pressure on him slowly begins to crush him, and at the end... he grows. But he takes a long hard road getting there. In this sense, the ML remains kind of static through a lot of the show, his growth doesn't really progress AS the show progresses. What you see is the buildup that leads to the changes he has to make, and how that affects him.
The Smile Has Left Your Eyes & Sora Kara Furu Ichioku no Hoshi - 6 & 7 respectively - The Japanese version has a lot more of the psychology and shows a lot more of the growth and struggle of the ML, but they are both good.
Tempted - 6 - A korean remake of Dangerous Liasons but only with rich kids. Although the plot is supposed to classically be about a personal breakdown of an ML having a moral crisis of character, this version is very unsophisticated.