Well, to a degree "more than surviving combat" is what you put into it... But here's some good ones I own and have read or used.
All Flesh Must Be Eaten uses the Unisystem like most of Eden Studio's games; it has a large number of supplements for everything from kung fu zombies to wild west zombies. It's a toolkit game where you can make the zombies work in a variety of different ways. It's the biggest zombie RPG. They are ruthless, I see that there's a WWII (Band of Zombies) and a pirate sourcebook (Arrgh, Thar Be Zombies) coming out this year. So it's well in print. I've played many one-shots of AFMBE and it's a good game. It's crunchier than most indie games but very low crunch compared to the GURPS/D&D/etc. set. But it's pretty trad in that relationships and whatnot aren't really modeled integrally in the system.
(Naturally you can run zombie survival horror in anything from D&D to, well, anything and someone probably has...)
There's a Savage Worlds based game called "War of the Dead," it's sold in some kind of chapter by chapter format that I don't understand. Zombie Run was a good earlier Savage Worlds setting, it focused on the moving cross country aspect that movies like Zombieland and parts of other movies use. I have an adventure called "Weekend Warriors" that is a military vs zombies scenario for SW too. You might choose this game if you already like Savage Worlds in other venues, it's a light trad game.
In terms of deeper story, there's a number of indie games I own that bring some additional dimensions to zombie survival horror.
There's a new zombie survival horror game by John Wick called The Shotgun Diaries that was up for an Indie RPG award. It's been reviewed glowingly from various sources. It's very short (18 pages), not for someone that wants loads of rules support.
There was an interesting zombie game that got runner-up in the 2007 Indie RPG awards called simply "The Dead," whose tagline is "A role-playing game about death and relationships." It is free for download. It focuses on relationships between survivors and how those give people each other strength (or weakness). I downloaded it at the time and was impressed, it's short (32 pages) but the relationship aspect is really neat.
There's a new post-apocalyptic game by Vincent Baker called "Apocalypse World" that's like 300 pages and well spoken of. You'd have to add zombies, though someone is working on a hack for it called "Dead Weight" that does so already... It's got a lot of support right now in indie circles (even if some if it is a little on the rabid fanboy side).
Adding zombies tends to be easy. I'd first go for what kind of apocalypse you want. A Twilight:2000 kind of WWIII military action thing with crunchy rules? A Road Warrior type like Atomic Highway? And then toss in zombies.
There's Left 4 Dead hacks for Savage Worlds, nWoD, and even 3:16 out there too.
Heck, I have a couple zombie-RPG PDFs I don't even know how I got... The RPGnow Haiti relief bundle maybe... Zombocalypse, Contagion... It's one of the most covered genres in gaming. Do you want lots of support or just good one-shot fodder? How much system do you want?
You'll need to make some setting adjustments, but not as many as I first thought.
Location: You'll have to tweak this a bit, but not much. It's set in a fictional version of early Mormon Utah, meaning it's actually quite a bit like the European Dark Ages: towns are isolated, the space between is wild, and religion is the glue that binds the people together.
Premise: I'm not sure this is what you had in mind, but I think it's appropriate for your goals. Players are "Dogs," traveling representatives of the Church, which means they have full and total responsibility and power: their job is to maintain the peace and keep "demons" (it's up to the party whether demons are literal or metaphorical in their campaign) from turning people to sin. Because sin is contagious, prevention is crucial and if it takes hold, the situation must be dealt with swiftly. Dogs may use any means necessary to do this... and how they exercise this power is the crux of the game.
Horror: In their battle against demons, the Dogs find themselves faced with distasteful moral dilemmas which only they can resolve --and they have no moral guidance; the Church will support any decision a Dog makes in the field. It is the GM's role to present Dogs with problems to which no solution is entirely moral: players are forced to choose what they feel is the best of bad options. The game explores whether the ends justify the means, and what impact this kind of choice has on those who make it. When a player makes a decision ("In X situation, I'll kill a man if I have to"), the GM then raises the stakes with a future scenario ("What if it's a kid, not a man?" or "What if the entire town is watching?"). Because these are weighty moral problems, players can feel the burden as well as the PCs, and the tension escalates as the GM raises the stakes.
Cooperation, tension, and half-winning: Dogs frequently fight amongst themselves because of differing ideas about how to resolve an issue; again because these are weighty moral problems, tension can run high both in and out of character. Cooperating in a conflict can provide a great advantage, but a good Dogs GM will be looking for ways to make success more uncomfortable than failure (in failure, at least you know the horrific consequences aren't the direct result of your choices).
Mechanics
Characters are built with stats (Acuity, Body, Heart, Will) and short phrases describing items, skills, relationships, and so forth. Each stat and phrase is associated with a die expression: "Acuity 4d6," "I'm a good shot 2d8," "my older brother whom I worship 1d4."
In a conflict, you roll the dice associated with every stat or phrase appropriate for the scenario. The die results are individually used in a betting-style resolution mechanic: one player narrates a declaration and puts forward dice to support it, then each other player put forward dice of matching or greater value and narrate accordingly. If you're unable to put forth enough dice, you're out of the conflict. The more dice you need to use in order to match the original number, the worse off you are (narrate it accordingly, and at the end of the conflict you take damage based on how many extra dice you used) but you're staying in the fight.
You can escalate to get more dice: taking a swing at someone who was just talking lets you immediately roll any associated dice and add them to the pool you can bet from, but as you escalate the consequences for losing become increasingly dire. Drawing a gun easily results in somebody dying.
The mechanics take a little getting used to, but they're so narrative that you can get wibbly-wobbly with them if you like.
In conclusion
Dogs in the Vineyard doesn't fit your setting, but it easily could. Your PCs would have to be figures of authority, but that's actually good: authority make it a lot easier to be tempted and to explore the darker urges of their nature. The game's base premise is similar to yours, as it focuses on the internal turmoil arising from dealing with external forces of evil. Consequences of failure (and success) can be as personal or far-ranging as the GM and players can imagine, and the mechanics provide great narrative freedom while still encouraging that darkness by offering extra dice as a reward for escalation.
I should mention that the game's design leads to episodic plot: Dogs move from town to town solving problems, resulting in a monster-of-the-week feel to the stories. Consistency in the campaign comes from the Dogs themselves as they explore (with not-so-subtle prompting from the GM in the form of thematically appropriate problems to be solved in each town) their own moral boundaries in pursuit of their righteous tasks.
Best Answer
Zombie apocalypses range from a slug feast (Left4Dead2) to a vehicle for huis-clos building on the survivors' slow but assured spiral into inhumanity (Walking Dead). It seems that you are aiming more towards the latter.
For me, Night of the Living Dead and Walking Dead and ... are all scary because they are all about the slow, methodical, and inextricable dehumanisation of their protagonists. The zombies, as is often the case, were nothing more than the catalyst for this. But could not have been replaced with anything with intelligence: you cannot reason with them, just get eaten.
First, alienation is a must. You should make sure that the players (yes, I mean the players, not their characters) are not comfortable. Stress them a little. Then stress them more. They should translate that into their character play. Setting, music, lighting, food, etc... Use it all! Clearly, get players' agreement before messing them! I do in all my horror games and provided I stay clear of specific trauma (as determined by players, not me), it works really well.
Second, let the players plot their own decent into becoming monsters. Or when you create the characters, make sure there is a path outlined for them to follow. The cop becomes a bully. The soldier becomes a murdering machine. The prom queen puts herself above all others... Stereotypes can be exploited here as it is a one off. Or do not go there. Maybe the mild mannered school teacher will turn psycho. I would let the players plot their own character's doom. It will add to the whole hopelessness.
Third, start slow and pick up the pace. The external Zombie threat must continue, like the rising tide, and never ever stop. Any gain the characters make, comes at a cost. More often than not, at a greater cost than it was worth. I suggest you have NPCs that you can sacrifice during the start of the game. Just make sure that each character has connections to them. Also, if (read: when) a payer character dies, make sure you have a backup.