Heavy Rain is a must. Telltale clearly drew some influence from it when making The Walking Dead. You might also like LA Noire.
What else...well, I guess any WRPG fits the bill. The Elder Scrolls, Fallout, KOTOR, Mass Eff...wait, scratch that last one.
Deus Ex has a lot of focus on choice too. You said you didn't like shooter games, though, and while Deus Ex is more RPG than FPS, there's still a lot of FPS elements to it. If you do get it, though, remember this: The original and Human Revolution are good, but avoid Invisible War like the plague.
You might also want to try some visual novels. They're kind of light on gameplay (in fact, whether they count as games at all is debatable), but if you want your choices to affect the story I'd say they're worth a shot.