I'm surprised no one has mentioned The Last Unicorn!
"Why do you come to me now?" had me tearing up, and the ending on the beach, with Prince Lear awakening, saw tears rolling down my cheeks. Even just listening to the opening theme makes me tear up, actually. This, very recently, despite seeing it about 50 times as a kid.
Also, despite efforts to maintain control, I cried 3 times during UP. Could have cried even more, if I'd let myself!
As a teen I cried during the Anastasia cartoon, which is pretty ridiculous, but... I was in a sensitive phase. Yes, that's it.