I wrote:
BUT... I have discovered that the Internet security device on the network is also running a DHCP server. My guess is that it is interfering with the process, possibly responding with an offer for the old address before the PC sends DHCPREQUEST. A classic race condition.
I don't manage that device so I'll have to request the necessary change. If things start working after that, I'll post a followup.
Sorry for the delayed followup. Fixing the security device did indeed fix the problem. It was never a problem with the Centos 6 DHCP.