I'd go for the baton. The axe is certainly more effective, but lacking the proficiency, I'm sure I'd end up hacking myself in the foot with it. And the last thing you want in such a situation is a nasty flesh wound from an axe with zombie blood all over it
I would choose the axe. Firstly, I would sharpen it and upgrade it till it becomes the best splitting axe. After that, I'll sit tight and be ready to confront the zombie apocalypse. On top of that, since I love books, movies, and tv series related to supernatural creatures, I think I'm pretty prepared to face a zombie apocalypse with everything I already know. I can't be that bad. Since I have enough food to make me happy and enough water to keep my body hydrated, everything it's fine, and I can get over anything.