I am into emotional storytelling as well, I think what gets you involved most are well drawn characters.
I would recommend planescape torment for an epic adventure full of stories, characters and heartbreak.
Really good stories/characters are few and far between in computer games but “The longest journey” comes to mind and even the first Witcher game.
Unavowed comes to mind as well as a rather affecting story.
And a couple of dozen books, but that’s not what you asked for…
I always loved the Song of Ice and Fire books by George Martin, be aware that the series is not finished, though. I am a big fan of Stephen King as well, he writes great characters. It, Shawshank Redemption, The Dark Tower… Thuesdays with Morrie by Mitch Albom is very emotional, I could not even finish it. Deadhouse Gates by Stephen Erikson. The Liveship Traders by Robin Hobb just to name a few…