I hope you have a DS, as it has some of the medium's best in this field. That is to say, the Phoenix Wright games, the Professor Layton games, Hotel Dusk: Room 215, and freaking Elite Beat Agents. All amazing games.
I also recommend Heavy Rain; though it has plenty of violence, it's not all centered around it and the gameplay is in no way built around it. It's also one of the best story-based games out there (and yes, it's a game; anyone who says it's too much like a movie has apparently seen a movie where their choices dictate the ending).
Does Puzzle Quest: Challenge of the Warlords count? You are fighting things, but the battle system is a Bejeweled-esque puzzle game. And it's freaking awesome.
EDIT: Also, Clannad. But it's tough to get a hold of; ask any visual novel fans here on the site. But it will make you cry, absolutely guaranteed. Clannad makes everyone cry. Clannad makes rocks cry. Clannad made me cry (and I never cried in shows or movies before then) then continued to gloat about how much it broke me by making me cry at everything since; seriously, I cry in everything now because of that show. Though admittedly, it's easier to get a hold of the anime than the original game. Maybe just go with that, but know it's based on a game.
I also recommend Heavy Rain; though it has plenty of violence, it's not all centered around it and the gameplay is in no way built around it. It's also one of the best story-based games out there (and yes, it's a game; anyone who says it's too much like a movie has apparently seen a movie where their choices dictate the ending).
Does Puzzle Quest: Challenge of the Warlords count? You are fighting things, but the battle system is a Bejeweled-esque puzzle game. And it's freaking awesome.
EDIT: Also, Clannad. But it's tough to get a hold of; ask any visual novel fans here on the site. But it will make you cry, absolutely guaranteed. Clannad makes everyone cry. Clannad makes rocks cry. Clannad made me cry (and I never cried in shows or movies before then) then continued to gloat about how much it broke me by making me cry at everything since; seriously, I cry in everything now because of that show. Though admittedly, it's easier to get a hold of the anime than the original game. Maybe just go with that, but know it's based on a game.