When I used to shave down to a clean face my lips would get chapped too. I never got razor burn or anything like that but my lips would dry up like the Sahara no matter what time of year it was. In the winter it was worse.
I used Carmex just to keep some moisture in but later I just said 'screw it' and stopped using a razor altogether. Now, no more chapped lips and I just use clippers to keep the stuff around my mouth tidy so I can still see them. LOL otherwise if it gets out of hand, everything I put in my mouth gets hung up in the hair and it can get smelly in a real hurry.