You should really switch to an electric shaver. Also, when showering try to lower the head to the point where your head is above the shower head and just lean back to wash your hair (if it needs it!) I just went 5 days without washing my hair because I didn't sweat. Your face might be oily at first but just don't touch that and let it soak into your skin overtime. Then your acid mantle will start to regain itself and you'll be thankful. Also what day are you on?
Have you been putting any type of water on your face? If you are gonna go caveman then I say go all the way and literally don't do anything to your face. Don't exfoliate or 'splash' with cold water at least until you are at least a month into this regimen. The only thing that touches your face is the pillow at night. Also, avoid water contact when swimming or showering!
Stop obsessing over which miracle product will cure you. None will. Take the lazy method and forget about your face. Do you think any of your clear friends stare at the mirror using ten different products on their face? No. Trust me, it might sounds crazy but it proves results. Dont put anything on your face. Once you get past the first few days its easy sailing from there. Google the caveman regimen