I recommend the drama
"Beloved" / Dear You.
You'll find that it's actually very hard to find dramas about actual married people unless it involves some really crazy things like magic, body-swapping or something. Usually the wedding is the dramas' finale. There are plenty of dramas about engaged people.
It's actually easier to find Japanese or Taiwanese dramas about married people instead of Korean dramas.
A really good drama about married people is
"Dare Yori mo Mama wo Aisu" (I love Mama more than anyone). It's very heartwarming.