Does Guangzhou Baiyun Airport have shower fecilities?

Yes! You can take a shower at the VIP lounge on 3F, East-I Bridge. However, it is just for first class passengers only.
Thus you are suggested to stay at the Pullman Hotel. It's located to the opposite of Gate 13, Departure Hall. The hotel provides comfortable environment and nice service. You will certainly take a shower and have a good rest there.
By the way, you are eligible to the 24h visa-free policy. Upon landing, just apply for a stay permit, and you can leave for the hotel. Have a nice trip! :)