The answer starts from the thinking man's response to zombies presented by John Ringo here.
Zombies, fundamentally are dumb. (or if they're not, we lose.) Attracted by noise, etc.
The first trick is to ensure that the characters don't know it's a zombie game. Instead, they're, for example, police, investigating an IED in a cemetery with a remarkable amount of carnage. A group has read Ringo's note and functionally deployed it against a small outbreak of zombies (though the fact that they were already dead at the time of the explosion wouldn't be apparent yet.)
This demonstrates that zombies can be easily dispatched if the players don't fall into the survival horror mindset and that NPCs are already doing it. The first couple investigations will be these "domestic terrorist" incidents, which gradually reveal "oh no, zombies!"
Of course, these "domestic terrorists" (read: survivors) are keeping the outbreak pretty well contained, and stigmergic communication suggests these techniques to groups around the country... who have to deal with zombie outbreaks in their cities. The players, being the first cops on the scene are drafted/pronounced experts/from a special group/PlotReason into investigating the various outbreaks, which expands into "Who did this?"
The moment you threaten the players with zombies and no resources... they'll go back to the horror mindset. Keeping the players as government coming in during/after the fact and trying to figure out what's going on as well as spinning it for the media: that's where you'll get the gumshoe and zombies game.
Here's a rough 5 act structure. The following scenario is designed to give stealable ideas for your game, in the sense of fleshing out the ideas above. It shows how the zombies are kept at one remove, and players are given time to observe, orient, decide, and act. They should know zombies are out there and have real resources to draw on before any zombie threat shows up.
This is one example of how that could work, the structure itself is roughly a 5 room dungeon:
Domestic Terror, IEDs, and Graveyards.
An Investigation in 5 parts.
Act I: Improvised Explosive Devices ... in the graveyard?
Characters are paged with a "get here now, there was what sounded like a huge firefight and the entire graveyard's on fire."
- Complex device at the center of graveyard, characterized as IED, solidly built, attack must have been planned for years
- Possible gang warfare as there are too many burned and charred bodies above ground.
- Graves have been disturbed
- Some bodies have flesh trapped in their teeth
- Way too many older bodies above ground
- Older bodies look like they've been moved in rings around the device
- Device has a "lightshow" function which displays bright moving lights, sounds, and what sound like screams.
Act II: The meat-grinder
Garbage collectors call in emergency call when they find dumpsters full of ... blood? And chemicals sprayed all around. Biohazard!
- DVD with a "how to protect yourself from the coming zombie apocalypse" on the cover, experts suspect that it's a terrorist coverup.
- More sophisticated IEDs, including a lure above a huge meatgrinder.
- Inside the structure, a "normal" person shot through the head, only other visible injury is a large bite on forearm. Body has had eyes closed and moved after death.
- Following trail of bodies back to another cemetery, signs of a ritual being performed
- Person who made the ritual watching Bruce Campbell movies on DVD player before hand
- [lots of fun details about both sides]
Act III: Wait, zombies?
Autopsy of corpse. Corpse gets up during the middle of the autopsy. players shoot it down (or room goes up in flames. Remember, threatening players with danger = they go into survival horror mode. They've got resources on their side. They've got the military on their side.)
- Corpse has strange [vector of infection] in blood stream.
- Belonged to registered survivalist group in [location]
- Survivalist group has published "How to survive the zombie apocalypse"
- Records for the group
- [more support for vector of infection.]
- [more details of the survivalist group.]
Intermezzo: The Clubhouse
Agents show up at group's clubhouse, asked some non-sensical questions, allowed inside. Group explains what they know about [vector of infection] and proof of the uprisings.
Act IV: Showdown
With evidence for the [vector of infection] the agents head to [badGuyLocation] to arrest the person behind this. Have to shoot through house full of zombies. Action-horror rather than survival horror. Let them have plenty of blam and prep. Allow them to indulge in creative IEDs demonstrated in first 2 acts.
- A pamphlet: "So you want to raise the dead..."
Act V: Zombie Publishing Warehouse
Warehouse, small company, everything normal. When agents arrive there's a scream from inside, and a mail-truck turning around the corner. Small lab inside, the inventor has infected themselves with [vector of infection]. Small shootout, should be no trouble if the group closes the door to the room first, then secures the room. The zombie threat has been neutralized ... right?
- Mailing list for people who ordered "so you want to raise the dead."
The essence of this is that the characters are functioning as investigators first and survivors second. They are approaching the problem with resources investigators can bring to bear: they have overwhelming force if they want it, they have resources, soldiers, and support. They are the cavalry. They are at one remove from the zombies, and can view the problem in a detached light. By giving them time, distance, and resources, you don't break their OODA loops and trigger "Oh crap! Survive!" responses. For most of the scenario above, they are reacting to reactions to zombies, rather than zombies themselves. By presenting evidence of zombies, without the zombies, the players can indeed arm and prepare against a described threat, neatly neutering any horror or panic that the players feel.
My other recommendation starts with a quote from the Esoterrorists's book:
In a horror game, mayhem suffered by the protagonists is not an impediment to forward movement. Horror characters are expected to die early and often. The player creates a new
character, the Ordo Veritatis ships him to the scene, and bingo, you’ve got more meat for the grinder and the story continues.
Make it very clear that the players belong to an organization and that their observations will survive. If some fall to zombies, reinforcements show up next scene and play continues. That way, characters themselves are not scarce resources.