Non-Romance drama but with couple as main leads

• Sousa Kaigi wa Living de! - Family/Mystery. Leads are married
• Big Mouth - Action/thriller. Leads are married
• My Name  - Action/thriller.  FL later dates SML
• Family Matters - Action/thriller. ML and FL are married
• The Aobas' Dining Table - Life/Drama. SML and SFL are dating
• Saitama no Host - Business/Comedy. FL and ML start dating near the end
• Yume de Mita Ano Ko no Tame ni - Mystery/Thriller. I don't believe this one was confirmed but ML and FL appear to be dating
• Wisher - ^ditto
• Eternal Brotherhood - Historical/Fantasy. Each ML has a love interest (one of which is FL)

⬆ Most are already established couples so the 'romance' is little to none. 

• Gyeongseong Creature - Action/Thriller. Focus mostly on fighting and survival 
• Summer Strike - Life/Drama. Focus mostly on FL's healing 
• Assassin Academy - Action/Mystery. Focus mostly on assassin work and mystery
• Secret of Three Kingdoms - Political/War. Focus mostly on survival and politics 
• The Imperial Coroner - Mystery/Medical. Focus mostly on investigation and friendship
• Lie After Lie - Life/Melodrama. Focus mostly on FL trying to get her daughter back 
• Kuragehime - Life/Comedy. Focus mostly on FL's friends 

⬆ A little bit more romance focused, like it's a relevant secondary arc but it's not the main focus. 

I'd recommend checking out the Slight Romance tag, if you haven't already ^-^ 

Misty