1. FL falls in love with ML, but at the very beginning of the drama he has a girlfriend.  Nevertheless, FL and ML are together.

2. FL falls in love with ML first.  She confesses to him, but he reject her and hurts her feelings.  She tries to fall out of love, forget about him, he falls in love with her.  (They are supposed to be together!) (It would be best for ML to apologize for hurting her feelings, but I doubt that such a scene would exist)

First Scenario

  • Sprout

2nd scenario:

  • Another Oh Haeyoung
  • A Poem a Day

More Than Friends (2020) fits your second scenario to the T (and yes, ML apologizes for being insensitive towards her feelings for him)

 Yuki31:
1. FL falls in love with ML, but at the very beginning of the drama he has a girlfriend.  Nevertheless, FL and ML are together.

1. FL falls in love with ML, but at the very beginning of the drama he has a girlfriend weird first love thing going on.  Nevertheless, FL and ML are together.

 Yuki31:
2. FL falls in love with ML first.  She confesses to him, but he reject her and hurts her feelings.  She tries to fall out of love, forget about him, he falls in love with her.  (They are supposed to be together!) (It would be best for ML to apologize for hurting her feelings, but I doubt that such a scene would exist)
 Yuki31:

1. FL falls in love with ML, but at the very beginning of the drama he has a girlfriend.  Nevertheless, FL and ML are together.

2. FL falls in love with ML first.  She confesses to him, but he reject her and hurts her feelings.  She tries to fall out of love, forget about him, he falls in love with her.  (They are supposed to be together!) (It would be best for ML to apologize for hurting her feelings, but I doubt that such a scene would exist)

  • Heartstrings

The ML doesn't have a girlfriend but he is in love with another girl at first. FL likes him. They end up together at the very end I think.

 Yuki31:

1. FL falls in love with ML, but at the very beginning of the drama he has a girlfriend.  Nevertheless, FL and ML are together.

2. FL falls in love with ML first.  She confesses to him, but he reject her and hurts her feelings.  She tries to fall out of love, forget about him, he falls in love with her.  (They are supposed to be together!) (It would be best for ML to apologize for hurting her feelings, but I doubt that such a scene would exist)

About the second request, if you don't mind a drama from 2011, there is New Tales of Gisaeng. It's quite the essential of the ML that begs FL for forgiveness. There is more than one scene of him apologising and being sorry about how he acted; a lot of angst... even too much maybe (?)     Of course, it's melodrama style, so be prepared to endure yourself or to skip some scenes :')