Jugglers (2017)

FL works for ML
ML changes Demeanour for female lead
Cohabitation
Secret from colleagues
Loyal ML
Very similar vibes

Both start off with two people with totally different believes, lifestyles and overal world views. After their strange first meeting in which the male lead takes a notices of the female lead (and he seems to be the first one quite smitten with her) the main leads start interacting at their workplace.
Both main leads start living together (in the same building, but different floors) and due to their communication solve all of their problems.
Both dramas have good communication between the mains (mainly, but not only throught conversations).

ML takes over the museum where FL is curator, there is a lot of misunderstandings between them in first few episodes but ml starts to fall in love with fl.