Melee is not really overpowered, IMO. If gun ammo was more common (or more accurately, it was easier to match guns with correct ammo) or could be manufactured/purchased then there would be no reason to ever touch melee.
Melee is silent, guns are not. Moreover, melee as it is now allows to kill very easily things that are not supposed to be killed easily (eg Hulks). I think, just like you can't kill a Hulk with a .22, you shouldn't be able to kill it with a crowbar, either. To kill the larger zombies requires heavier ordinance (most commonly a couple of shotgun slugs. Which makes killing them noisy). Likewise, they shouldn't be killable by normal melee weapons (should require at the very least a katana or a machete), and should have drawbacks (at the very least no melee stun).
Basically anything melee does, guns do from range and with 0 skill required. Try killing a giant wasp with an axe with 0 skill? You're dead. Handgun? 1 - 2 shots, it's dead before it's even in range. Even with skill, there's a chance you don't 1-shot the wasp and it gets to sting you with the melee weapon.
You do require skill, if you intend to hit anything beyond melee range.
I think the whole thing as it stands makes melee a no-brainer for characters. That at 0 skill guns are better is irrelevant. It's so good, and training it is so easy (so much easier than gun skills, in which you're limited by their noisiness, and the lack of ammo) that makes all other combat skills pale in comparison.