Drama that includes marriage.

Of course you have Full House (they married a few eps into the series but they weren't "together". Korean, then there is the Thai remake and soon another one will be out chinese)
Delightful Girl Choon Hyang
Goong
Golden Bride - I think this one is the most similar to what you want, also was popular at the time. (*looks at airing date. dam do I feel old knowing about this one*)

Sort of (married half way or to another person): Bride of the Century, Que Sera, Sera.
I think this topic might be more common is weekend dramas (which look to me like sitcoms) since they are more about family and such.
Also Fated to love you and Itazura na Kiss there are different versions of them. INK there is the Taiwanese version It Started With A Kiss (I think that has 3 seasons) and an a lot older Japanese version (like from about 2000), Fated To Love you also was made in Taiwanese version before the Korean one.

One More Happy Ending! It doesn't emphasize the marriage life but the drama tackles different problems regarding marriage like remarriage, divorce, not being marriage, etc... you should check it out if you feel like it! :) 

I like this theme, but it's not popular in Asian dramas, it seems. Mostly all the shows about marriage life are negative. Focused on cheating, people staying together even though they should clearly divorce etc.

Closest you'll come to a drama with a more positive & optimistic outlook on marriage is Golden Bride, which I see has already been mentioned. My guess would be because after marriage people in Asia think the good times are over and life becomes all about fighting, surviving, hating on each other, cheating and oh yeah, the children. No more romance, lovey dovey stuff.

Western shows are more optimistic about love ... and love after marriage ... than Asian ones.

If you want a show in which the leads co-habit (they had a good reason to), check out Too Late to Say I Love You.

Bride of the Century (2014)
Main actors: Lee Hong Ki and Yang Jin Sung


Goong (2006)
Main actors: Yoon Eun Hye, Kim Jeong Hoon, Song Ji Hyo, Joo Ji Hoon

This is my favorite trope, so I have to say while they aren't that popular in Korea, you can find them a lot in different daily dramas, weekend dramas, that sort of thing. For example,My Daughter Seo Young, Unexpected You are just a few examples.

Right now I'm watching Marriage Contract with Uee, but the marriage is a hidden one, and there's also 1% of  Anything, in which couple gets married about halfway? through show. 

For non Korean shows, Thailand has a lot of them featuring arranged marriages, contract marriages, etc. I just finished Padiwaradda which was all about marriage and what makes the best wife and so on. 

I agree with @TheTranquilKnight's choice. A slice of life drama which focus on marriage, children, family life with a positive approach is rare in dramaland. You will find what you're looking for in 'Kimi no te ga Sasayaite Iru'. It's become a firm favourite of mine because marriage theme is something I enjoy watching and this one feels so naturally warm and romantic, caring and supportive. I'm happy to have found this little gem because I don't think I will find anything else like it (and I've searched for a long time!). I also loved the anime series Itazura na Kiss.

The Kdrama "Wedding" focuses on married life, but it older (2005).