The oldest drama I ever watched is Aishiteiru to Ittekure a japanese drama from 1995 (the Golden time of Japan when it comes to romance dramas, that was the 90s). Its a simple love story between an woman and a deaf man :) This drama has a rather mature feel with lots of skinship and real kisses! The first 9 episodes (there are 12 episodes in total) is gold ~
If you don't mind tragedy then I recommend the drama Kamisama Mou Sukoshi Dake (1998) a touching beautiful love story between an high school girl that has been diagnosed by HIV/AIDS and an famous composer. It's very sad but a great watch neverthless! Real kisses and skinship :)
MARS is a taiwanese drama from early 2000s, about a very shy / nice girl and a badboy/womanizer who both has dark pasts and are deeply wounded... but they come to love and heal each other. One of my all time favorites! The chemistry here is amaaazing because the actors were even a IRL life couple at that time, lots of real kisses and great skinship here as well!
I heard of Dung Duang Haruetai, but I heard the '96 version has not much romance , is that true? Is it still worth watching? Did you know that there is a new remake that will air soon with Tik and Kimberley :)