I really love dramas where the couple is together happily and the central conflict isn't really their relationship,  for example, Strong Girl Do Bong Soon and Shopping King Louis. I don't mind if there is a bit of jealously like in SGDBS but I don't like the whole misunderstanding trope (unless it's more of the super funny light hearted kind).  

So I'm looking for some recommendations of romance dramas like those two, not in terms of their plot but really the state of the couple's relationship throughout yet still has a really engaging plot.

Meeting You - there is almost no conflict at all. Some misunderstanding are present, but not between main couple. It is slow slice-of-life romcom cdrama (from friends to lovers and from high school to college).

You Who Came from the Stars - funny interesting romance kdrama. The main conflict is about something else than romance between leads. I don't know what else to write without giving major spoilers... :)

Shanai Marriage Honey - this short jdrama is one happy moment after another with a lot of skinship between leads. So if you want something funny, fluffy and with no conflicts - this is it.

I second You Who Came from the Stars - one of my all-time favorites. 

Another vote for You Who Came from the Stars - it was such a great drama and is one of my top three favorites!  

I thought the Chinese drama You Are My Hero was also really good and the main leads in that show have a very mature and healthy relationship.