Look into visual novels. That's the most abundant source of games similar to this. I'd play 999 and Virtue's Last Reward, and if you like those, try some others like Saya no Uta, Planetarian, Ever 17, and Muv Luv Alternative for example.
And while Heavy Rain and Indigo Prophecy are most similar to The Walking Dead gameplay-wise, the general consensus is that their writing is not really that great, and going straight from a very well written game like The Walking Dead to one of those will likely be fairly disappointing.
And I will also recommend Nier. While there's not much choice at all, (though there are 4 endings) I'd say in terms of the tone of the story, it is the closest game to The Walking Dead that I can think of. It's story and soundtrack is about as good as it gets.
This. Visual Novel (VN) games are quite easily the closest thing to The Walking Dead, so much so that many even consider TWD to be a VN game itself (which is up for debate, but that's another topic altogether).
Anway, the OP should definitely check out the VN games listed above: the Zero Escape series (999 and Virtue's Last Reward), Ever 17 (predecessor of Zero Escape series), Saya no Uta, Planetarian, Muv-Luv Alternative, etc.
In addition, another VN I'd add to that list is Clannad. The whole father-daughter-like relationship in TWD reminded me of Clannad, especially the After Story portion (i.e. second half). Anyway, Clannad had some of the most emotionally powerful drama I've seen in any medium, so I'd highly recommend it.
And oh yeah, if the Zero Escape games interest the OP, then I'd also recommend checking out the Infinity series, which includes Ever 17 (mentioned above) and Remember 11. Another VN I'd highly recommend is Steins Gate, which has one of the most engaging and thought-provoking sci-fi narratives I've seen in years, in any medium.
And beyond VN games, the OP might also want to check out some other VN-like adventure games, such as Catherine and To the Moon, both of which arguably had the best video game narratives for the previous year of 2011.
Thanks! Catherine has been on my radar for quite a while now. Think I'll be picking it up along with 999 during my next game drought
Good choices. I'd say 999 and Catherine (as well as The Walking Dead of course) are great introductions to the Visual Novel genre.
However, as has already been pointed out above, some people didn't really like the puzzle sections of Catherine, though I thought they were pretty good (and even better in multi-player).
And oh yeah, there's one more VN-like adventure game I forgot to mention: Shadow of Memories.... if only because it was the game that introduced me to the VN style of branching narratives back in 2001 (many years before I knew what a VN is). It also has a VN successor called Time Hollow, though I've never played it.