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No, leaving zones while in battle would be too easy, you could just run past all the zombies without a single fight. Also the system can't handle it. So even if I wanted to make it possible to leave zones while in battle, I couldn't.

You can heal up during a fight, if you had any items set up on your action bar. If you followed the tutorial, you were instructed to equip Small Medicine.

If you run away from zombies, they will give up the fight, they retreat to their original spawning point and you can leave the zone. But you're not supposed to pass all the zombies without a fight. Specially not if you just run past them. I might even disable running for this same reason. You can make it to the other side without battle, if you're careful.

Zombies notify with a (!) when start attacking you. When you see a (!) above a zombie, you know it comes after you. 

I feel like you just want it to be easier, but... The plan wasn't to make an easy game. Getting passed zones should be a challenge. Staying alive is just a matter of focus. Use ranged spells (Maaike has one, and the main character gets one pretty early in the game, you also can find or buy ranged weapons) to avoid bites. Use your grenades when zombies lock you in... And bring the pizza with you. (There is one in the oven in your home.)

If one of your characters is about to die, you can switch characters in the menu. You have to be out of battle for this. You can indeed not open the menu during battle. You have to make sure your action bar and weapon wheel is setup before you start a battle.

The trick is to not stand still and fight as less as zombies at the same time as possible. If you just don't run into a group of zombies at once, you should be fine. Just don't stand still, and don't fight more than 1 zombie at the time. If you have to, throw a grenade.

The gun and others uncommon weapons are indeed not cheap, but they do drop regularly, check your inventory. Drops have semi-randomized stats (they can be 2 points stronger or weaker than the stats given in the database) So if you have 5 pocket knifes, there is probably one stronger than the others. 

Maybe good to know if you want a better weapon: scientists carry pockets knifes, guards carry bombs and ammo, and cops (undead ones) drop guns, ammo and army knifes. If you get a gun though, use your bullets wisely. Bullets are pretty rare to find, and are 9g each, or 99g for 12. Early in the game, you're better off spending your money on beer for AP (mana).

I hope this helps you survive longer, maybe even not die at all. Should be doable, I can do it too.