In Marriage Contract the ML and FL (as the title indicates) are in a contractual marriage and she hasn't completely gotten over her husband who died 7 (I think) years beforehand.
In Sound of the Desert, the ML and SL meet the FL in the desert on the same day and she tells each of them that she'll be going to the city where they're from later on. Both guys look for her, but the SL finds her first. She develops feelings for the SL and he develops feelings for her, but he rejects her advances because he doesn't think he can make her happy. At the same time the ML is also pursuing her. They never exactly "date" (it's a historical drama), but they get engaged.
The SL comes back into the picture at various points and she is not 100% over him, which hurts the ML, but she stays true to the ML and in the end they basically ride off into the sunset together.
I can't think of any dramas where the FL is hung up on her ex while she's dating the ML and the ex stays in her past, but I can recommend a few dramas that have somewhat similar dynamics.
In I Need Romance 3 the FL looked after the ML when they were kids and the ML had a crush on her. He moved abroad but promised to come back. When they're adults, he comes back and gets to the point of almost dating the FL
before revealing his identity, at which point she treats him like a younger brother.
They live together,
and the ML helps her realize she has feelings for her boss / mentor.
She eventually dates her boss.
All of this hurts the ML, who is very open with the FL that he likes her through all of this.
Finally she realizes her feelings for the ML and dumps her mentor guy.
Slightly different dynamic, but maybe try In Time With You. The ML has liked the FL since they were in high school, but out of fear of dating and then losing her, he told her that he would never have feelings for her. Over time they became best friends. The drama starts in their adult life (with flashbacks) and the ML hates the FL's asshole ex. When she mentions him, it hurts the ML.
She later ends up dating the ex again and that whole arc hurts him.
In the end she realizes her feelings for the ML and what a dick the other guy is.
In I Hear Your Voice the leads are also not dating, but they live together and the ML has liked her since they were kids. When she shows interest in and dates one of her co-workers, that hurts the ML. She denies her feelings for the ML because of their age difference (it's a noona romance), but she accepts his feelings in the end and isn't hung up on the co-worker.
In The Eternal Love, the soul of the FL (Xiao Tan) travels back in time and enters the body of a girl (Tan Er) who is in love with the 1st prince (and he likes her) but is engaged to the 8th prince / the ML. The girls have very different personalities, and their souls share Tan Er's body. After they get married, the ML falls in love with Xiao Tan and she falls in love with him, but Tan Er still isn't over the 1st prince which creates situations that hurt the ML. In the end Xiao Tan can focus solely on the 8th prince.
In The King's Woman, the King has liked + been looking for the FL since they met as children. When they meet again, she likes someone else. He blackmails her into agreeing to marry him and gives her three months. She goes back to the guy she likes and when she returns to the King she's carrying that guy's child. The King accepts the child as his and eventually manages to get the FL to fall for him, but the child (among other things) is a constant reminder of the other guy. She chooses to go back to the King even after escaping the palace, but she never 100% stops caring about the other guy.
Also it's complicated and difficult to explain, but Empress Ki kind of fits too.