Regarding food scarcity, I think you only need to shift the start from “the day after” to “a couple of weeks after”. I can come up with a few reasons. Maybe you were hiking with your buddy only to return two weeks later and the world has gone to shit. Maybe you woke up from a coma in a hospital, a la walking dead. Or maybe you have holed up in the evac shelter for weeks, waiting for rescue, but now you’re running out of food, and you need to get moving.
Without power most fresh food will have gone bad or be on the verge to, after a few weeks. The only thing left is canned and dried food, which would normally be quite abundant… but not when people have already looted and scavenged for weeks trying to survive.
No zombies I know eat any food that isn’t meat, anyhow. Not that you have to follow zombie lore; whatever; but I think food scarcity is conceivable without it.
As for cars: I think cars should be a common sight, but very rarely functional. I imagine highways filled with cars to the brim, corpses all around, and zombies as well. The cars are all rusted to shit because of the acid rain, but they may well contain some goodies if you can clear out the zombies! Maybe a few functional parts, maybe some food and clothes in the boot. Whatever isn’t fucked by the acid.