I don't think a short sword would be good for zombies. A sword is good for cutting and stabbing, but zombies aren't going to stop for that. You either have to smash their skull, or cut their head off to stop them. You just need more IMPACT that a sword can provide.
I would rather have some kind of mace or axe. That way if they don't die, at least the impact would throw them back. Of the two, I would rather have the mace. It won't get stuck in them.
On second thought, you can't cut off a head with a mace. Let's go with the axe. Just be careful not to get it stuck. I would pick an all-steel type. You don't want the handle breaking. Go shorter, for tight spaces, but make sure it's not too light. You want a good head smasher.
Here's the perfect one, an Estwing camp axe. It's super well made, one solid bar of forged steel. Heavy, but not too, and short enough to stick in your belt and not trip your knees up. You can also conceal it with a trench coat.
I sprayed mine with candy-red metallic paint.
The khopesh is a sword which is used much like a battle axe. Some scholarly theory even suggests the earliest versions of the khopesh from ancient mesopotamia were developed from the 'epsilon axe'.
As such the edge on a proper khopesh should be rather axe-like in profile. I hand convexed mine and it cuts like the dickens while still having the impact effect of an axe.