The FL is already in a relationship

• The King's Woman - The FL is in a relationship with the SML at the start of the drama but the ML first fell for her when they met as children. He definitely pursues her while she's with the other guy and basically kidnaps the FL and threatens the SML's life in order to force the FL to stay with him. The FL is pregnant with the SML's child at that point so there's definitely baggage.
• Before We Get Married - The FL is in a relationship at the start of the drama and the ML falls for her first, but the FL's relationship drags a lot and the ML is also in a (failing) long term relationship. The ML pursues the FL in a shallow and aggressive way at first and doesn't care about their attachments or her feelings. He later decides to take his feelings, her feelings, and pursuing her more seriously.
• Secret - The FL is in a relationship at the start of the drama, but it quickly goes bad (she takes the blame for the SML committing a hit and run and goes to jail in his place while pregnant with his child). The ML was in love with the person who was hit and is bent on revenge, but falls for the FL while exacting his revenge on her / while realizing the truth that the SML was responsible for the accident.
• Ms. Perfect - The FL is married at the beginning of the drama but her husband cheats on her  (he is absolutely awful) and their marriage is clearly falling apart. Even though they're obviously not going to stay together, the FL's husband and another woman play a bigger role in the plot than I would've preferred. The ML falls for and pursues the FL first.

Not really in a relationship, but something similar:

• Empress Ki - If I remember correctly, the King of Goryeo and the FL weren't in a relationship at the beginning of the drama, but they both liked each other. There's a bit of dragging. Part way through the drama the FL gives birth to his child. The ML falls for and pursues the FL first. 
• Her Private Life - The FL isn't in a relationship in the beginning, but is a committed fan of an idol. The ML also thinks the FL is in a relationship with her female best friend and that she has to hide the fact that she's in a same-sex relationship because of Korea's conservative values (and also the small detail that her best friend has a husband). When fans of the idol think that he's dating the FL and send her threats, the ML agrees to fake date her in order to protect her from the threats without exposing what he thinks is her "real" relationship. The ML falls for her while they're fake dating.
• Sound of the Desert - The FL isn't in a relationship at the beginning, but near the beginning of the drama (not right off the bat) she does live with the SML who she has feelings for and who also has feelings for her (although he later rejects her confession because he thinks he wouldn't be able to make her happy, which he regrets doing almost immediately). The ML falls for and pursues the FL first. The love triangle is a major part of the plot throughout the whole drama. 

Not at the very beginning of the drama, but:

• I Hear Your Voice - Not at the very beginning of the drama, but in the first half the FL briefly tries dating one of her coworkers. He's a nice guy and it doesn't drag too much.  
• The Smile Has Left Your Eyes - Not at the very beginning of the drama, but in the first half the FL tries dating one of her brother's coworkers. He's sweet but very awkward. It doesn't drag too much.
• The Story of Ming Lan - Not at the very beginning of the drama, but in the first half the FL is in a relationship (I can't really say "dating" because it's a historical drama, but they were basically dating) with the SML where they nearly get engaged. It drags a little bit, but I wouldn't say too much.
• Lucky's First Love - Not at the very beginning of the drama, but in the first half the FL tries dating for the first time. The ML has already fallen for her at that point but it forces him to acknowledge his jealousy. 
• Moon Lovers: Scarlet Heart Ryeo - Not at the very beginning of the drama, but in the first half the FL starts something (not quite an official relationship as the FL lived with the SML and the SML's wife (sort of her relative) while the SML's wife was dying) with the SML. It didn't drag very long. 
• Go Ho's Starry Night - The FL starts the drama relatively newly single, but we see flashbacks part way through the drama of the ML holding back despite his feelings for the FL in the past while she was in a relationship.

 emaureeng:

Nice Witch - twin plot. The FL is married, secretly taking the place of her ill stewardess sister. (Not a fantasy, despite the title - the stewardness sister is mean, the FL entirely too nice for her own good.)

Thank you for this rec. I added it to my watch list. BOnus is: I love Ryu Soo Young and have been searching for more dramas with him as the ML and not the 2nd Ml.

Chugakusei Nikki - FL is happly together with her boyfriend but she end up falling in love with the ML who is her student. He fell in love first tho.

Majo no Jouken - Almost the same inicial story as Chugakusei Nikki.

Hirugao - FL is married but not so happy. She fall in love with a genty and nice biology teacher. There is cheating.

Love Affairs in the Afternoon - Korean adaptation of Hirugao. 

A Good Wife  - FL is married with a workaholic man. She met a nice guy and end up falling in love with him.

A Wife’s Credentials - FL is marred with one kid. She end up in a relationship with her son's dentist. There is cheating.

Long Vacation - She was going to marry her BF but he left her in the first episode. So she starts living with another man (his BF roomate), and ends up falling in love with him.

Love Again -FL is married. In a class reunion she meets her crush whe she was in high school. They end up falling in love again.

On The Way To The Airport - She is married and has a dauther. She meets her daughter friend's father and something tragic happen. In the middle of all the situation they fall in love. There is cheating.

Secret Love Affair -She is married and fall in love with her music student. Cheating.

One Spring Night -FL has a boyfriend in the start of the drama and fall in love with ML.