I'll recommend some older dramas since lots of the dramas here are also what I had in mind.
Chinese and Taiwanese dramas:
1. Huan Zhu Ge Ge/Peal Princess/My Fair Princess Seasons 1& 2. Whether it's Er Kang and Zhi Wei or 5th Prince, Yong Qi and Xiao Yan Zi, both male leads fall for their respective love interests first.
I did not finish watching the 2011 remake. Can't comment on that.
2. Ghost Husband . Not in mydramalist. Really old drama.
3. Dreams Link 2007/Dream Behind the Curtain.. Both the Alex Fong and Zhang Jiani and Liu De Kai and Vivian Chen adaptations are good. Dream Behind the Curtain is the older version .. I think it was made in the 90s or late 80s. It's about a girl who has an unrequited love for her sister's boyfriend and then an older man offers to marry her.. and she only falls in love with him way after they get married.
4. The Fox's Summer. Modern drama.
5. Mars starring Vic Chou and Barbie Hsu. Modern drama.
6. Devil Beside You starring Mike He and Rainie Yang. Modern drama.
7. Summer's Desire.. Super angsty and melodramatic. New adaptation coming up starring Sophie Zhang Xueying and Qin Jun Jie. I didn't really like the one starring Barbie Hsu, Peter Ho and Huang Xiaoming so can't remember much except that HXM's hair was weird.. and that Peter Ho was controlling... but kinda hot?
Korean dramas
1. Moonlight Drawn By Clouds/Love In the Moonlight (historical)
2. Reply 1988 (modern drama.. oh but it's set in the 80s... and early 90s but it's still not a historical drama in the conventional sense I guess.)
3. Arang and the Magistrate
Any adaptation of Hana Yori Dango or Boys Over Flowers whether it's the Japanese, Korean, or Taiwanese.. the Chinese one starring Zheng Shuang was a bit draggy. And there's a new Chinese adaptation coming up starring Shen Yue.
Also do read this Dramabeans post on dramas where the heroes fall first.