Katawa Shoujo... I don't remember if i cryed but that is the only one i remember giving me the feelings for that.
Apart from that I do not think I have seen a single game with enough depth in a character and its relationship with the mc to be interested beyond the sexual side.
One of the only games that really interested me beyond the sexual side and led me to play for the story itself is I love daddy.
As I have rarely tried games completed, it does not help ^^
I imagine that a lot of Japanese games could offer such feelings but I do not have the time and the desire to look for and play old games.
On the other hand, some made me laugh by their stories or dialogue. (Not just because it's so rotten that it gets funny, for that there are plenty)