I have found the most adorable romance couples in long family melodramas and usually they are second characters. 
The Gentlemen of Wolgyesu Tailor Shop-Tae Jang and Hyo Won
All About my mom-Hyeong Sun and Jae Ri
The wife's lies- Dong Xu and Yian Juan 
**the most recent finds.
I would probably have to re watch all the dramas I have watched to make a list of all the adorable couples I Love.
I think I love this couples because both of characters are cute and usually one loves Aegyo mostly the girl and they are very open to kissing and hugging. They are open to show they love each other. The guy usually ignores her even to a point of threatening her to call the cops if she doesn't stop following him. She usually follows him around all thru the drama ( which gets on my nerves because she has no self love but is beautiful when they start to date hahah)  Their relationship is not based on passion but sweet romance. Take in count this couples are favorites on that category because i have list of best chemistry, the ones I ship more. 
This dramas I mention are one side love until the guy falls for the girl. The guys usually have a girlfriend they have to leave for whatever reason or the girlfriend breakups with them. The girls are rich the guy is poor. 
I love the chase not insta love and this couples has that. If you guys have recommendations on similar romance Let me know.
Please recommend me dramas with This type of couples it can be leads or secondary characters. It can be Korean, Japanese, Taiwanese, Chinese, or Thai. 
@IzzyDramaLover Thank you!!
any version of: Itazura na kiss (the only thing is that the girl is not rich and the guy is not poor, but it fulfills your other requirements)

Tatta Hitotsu no koi
A Gentleman's Dignity (Choi Yo On&Me Ah Ri)
Ugly Alert (Hyeon Seok&Shin Joo Yeong)
Seo Yeong, My Daughter (Sang Woo&Ho Jung)
Protect the Boss (Moo Won&Na Yoon)
Family's Honor (Hyun Kyu&Hye Joo)
My Girlfriend is a Gumiho 
I looove this kinda trope too.. Its very cute when the girl falls for the guy first when he has nothing to do with her cos it gets adorable when they start to date..
@fanitha thank you I have watch all the remakes and original. But I only liked the Thai version as the guy wasn't as a big of a douche bag as in the other versions. 
Will look for the other one. 
@sandy wow those are a lot thank you so much. 
@sandysan yeah I feel like the love is more unique as is a slow burn love.