Wonderful Fate (2021)

Modern day FL enters a historical fictional world, and falls in love with the antagonistic or villain ML.

Parallels between My Devine Emissary and Wonderful Fate include:
The female lead is a student who is struggling academically and is reluctant to take her tests.
She travels back in time and/or into a historical narrative, where she encounters the ML, the emperor.
The connection between the leads in both dramas is that the leads characters are quite similar in that they initially despise one another before becoming friends.

The dramas are sort of the reverse of each other - WF is a real world person entering a manga world, TMRFTS is a manga character entering the real world.

Fluffy light time travel romance with a happy ending after ML comes to present day. Quirky and fun FL and endearing ML.

In HL, a character from a manga enters the real world, WF is a person from the real world entering the world of a manga. They then fall in love with a person from that world while attempting to change the story.

FL enters the world of a story (novel vs. drama), initially views the ML as the villain. The ML distrusts her at first but slowly falls in love with her.

Both dramas have a modern day FL entering the world of a historical novel and falling in love with the villain or the character she originally hated.