Work WITH your players
If I'm going to sit down and play a game for several hours, nevermind for several sessions of several hours, I want to know what genre I'm playing. The "gotcha" campaign doesn't make people as happy as folks think.
Tell your players, "I'm planning on doing a zombie outbreak game, and the expectations are that your characters are going to be built like (Normal civilians/civilians with some survival skills/police military etc.) and the first part will be before the outbreak."
What's to stop players from doing stuff like building a bunker before play? Because you, as a group, agree that's not the kind of zombie game you want to play, and if they don't want that, they're not going to like this campaign.
Easy.
Motivation & Relationship Mechanics
This is one of those places having relationship and ideals based rewards would work great.
When a media wants to show you more of the pre-outbreak, they show the protagonist's normal lives - the people they care about, their ideals, hopes and dreams and struggles. In games with these kinds of reward mechanics, you spend these scenes loading up these relationships and ideals, getting either XP or some kind of hero points in the process, and then when all hell breaks loose, then the players get more points for trying to protect, rescue, or reconnect with those people, places, and things.
"The Univesity. You know, I would have been the first one to go to
college, Mom wanted that for me. And here I am. Finally walking
through these doors. But I didn't want it to be like this."
These kinds of mechanics work very well because they set up a long term survival goal into the rules - in the short term, doing what's immediately best for survival is good, but in the long term, you want those XP/hero points to stay alive as the situation worsens.
Which...mirrors a lot of zombie fiction - the last survivor(s) end up having to balance caring for others and being actually a decent human being vs. pragmatic "I'm sorry, you've been bitten, I can't let you in.".
Social Mechanics
Games with good systems for social mechanics would also be good. Zombie fiction is a lot about the arguments about what to do next and everyone getting at each others' throats and sometimes people making bad decisions. Good social mechanics gives people a chance to influence each other, to convince each other to help, or to cut bad deals.
Tying it together
When you have the two above ideas working together, you can use the pre-outbreak time to set up relationships, rivalries, tensions and promises. All of that becomes critical when the disaster strikes.
"I know he's your father. But he's been bit. Our time is running
out. He made you promise you'd keep yourself safe and try to live the
best you could, right? You won't be safe staying with what he's about
to become. Do you want to protect his corpse or his wishes?"
You'll notice all that kind of stuff is what makes the best zombie stories work. It's about people falling apart and falling together that makes it work, not "I am tactically sound and have food supplies" kind of stories.
There is no zombie formula. The 3 types of zombies featured in the Monster Manual aren't independently created creatures, but they are independent of each other.
The basic Zombie is one of the staple enemies of D&D, along with other swarming mooks such as Skeletons, Goblins, and Kobolds.
The Ogre Zombie is simply an interesting variant on the regular Ogre. Lower AC makes players feel more effective, with higher HP making sure the battle is no less challenging. If you look at the melee attack of the 2 creatures, you can see that it is identical, because they're effectively the same creature.
The Beholder Zombie is one of 4 types of Beholders. They all follow the same pattern, but with varying levels of power. The basic Beholder is a CR 13, with the Death Tyrant being a slightly more powerful version at CR 14, the Beholder Zombie a much weaker version at CR 5, and the Spectator bringing up the rear at CR 3.
Spectrums like this one are fairly common in the Monster Manual, particularly with iconic creatures like beholders. This allows a DM to include one of the classic monsters in their campaign regardless of whether it is a high- or low- level campaign. Other examples of this phenomenon include golems, hags, demons, devils, dragons, giants, angels, lycanthropes, and even oozes. (I could go on for longer, but I think you get the idea.)
If you want to zombify a creature, start by deciding whether you want being a zombie to make it more, less, or equally challenging. Then add zombie traits such as Undead Fortitude, lowered mental abilities, and increased Constitution. You can adjust HP and AC depending on how challenging you want it to be. You'll have to do this individually for each creature, though, if you want to maintain its unique flavour while adding zombie flavour. The DMG has instructions on how to recalculate CR once you finish modifying your creature.
Best Answer
The problem is that zombies are pretty strongly typecast into survival horror, so you have to do something which breaks the mold.
Something should happen which doesn't fit the typical zombie game... something unusual and mysterious, which prompts investigation.
The most basic idea is a cure. For example, make one character turn out to be immune, and have them backtrack to find out how this happened and if there's a cure. Use the cure as a kind of MacGuffin to lure them into finding out the source, and discard it as a plot device once they know the source of the outbreak.
Another possibility is the outbreak doesn't spread from person to person; it obviously comes from a specific source... rip a page out of Aliens: victims are either eaten completely (ie not rising) or they are carried off to some kind of nest where they are exposed to the contagion. Make it really obvious that the zombies are "choosing" people; have them sniffing and groping people before hey get carried away.
But the point is you have to break the zombie-survivor-horror mold; establish clearly that the zombies are a stylistic element in a different kind of game.