The problem appears to have been fixed. There was no one clear culprit. There was some kind of timeout occurring between my websites and CloudFlare and that caused the error messages.
The problem cleared up for me and wavesoftech after CloudFlare fixed an issue with IPv6 connectivity to http://www.cloudflare.com but they were not sure that was the root of the problem.
Nevertheless, I no longer have the error messages and I can change my settings in W3 Total Cache without getting error messages.
Only certain hosts were affected by this so this must have been a bad combination of specific host configurations with CloudFlare.