If you want the axe, its VERY easy to get.
1. At the start of the game, find out where the hospital is, and on the way to the hospital, find a shotgun, if you can't find any, detour to the nearest shopping district and get a shotgun. Also get a flashlight if you come across, but its not necessary. Also please bring some food if you want to survive.
2. Hurry to the hospital, the earlier you get there, the lower the chances that the folks in the hospital will get infected. If all the inhabitants of the hospital are infected, you might wanna start a new game, its just simply not worth the effort and ammo to get rid of the zombies in the hospital
3. Head down all to the corridor, get to the generator.
4. Now cautiously move around, Jason Myers should be somewhere, a flashlight helps, especially, to see a longer range.
5. When you see Jason Myers, RUN behind the fences and shotgun him, OR, if you are too far from the fences, run and shotgun him, shotgun almost never misses, and 3 shots will take down this madman. ALWAYS stay 1 at least 1 step away from Jason, he'll most likely insta kill you early in the game when you have low stats. Also if you think of meleeing please don't ever do that.
6. Turn on all the generators and head upstairs, the door is now open, get some loot. Also save before you head upstairs into the hallway after you've gotten your medkits.
Now this is highly dependent on the RNG, I had one run when I went down the hospital and everyone was alive, after turning on the generators and killing Jason Myers, I went up again to find EVERYONE infected. Lots of neophytes too.
If you encounter that, reload if you haven't prepared for the disaster. Your gonna need LOTS of medikits, also if your low on sleep, get some blue pills and eat them, you don't wanna be running through the hallways sleepy.
Now don't bother fighting, just RUN! If you have a wall of zombies blocking your way, try to step back a little, they might get out of formation. When you get to the top, leave the district unless you have a really good reason. That is because the zombies will pour out from the hospital and chase you.
Now get that axe and kite some zombies