I got the same issue to, just installed w3 total change and made the configuration on my 2003 server.
Unable to communicate with CloudFlare API: Setting: zone_settings SSL certificate problem: unable to get local issuer certificate.
But I do not thins this have anything to do with Wordpress. It's Cloudflare that have an issue with their ssl certificate, maybe it has expired or something. Hence the "unable to get local issuer" comment. The server cannot validate if its authentic or not and closes the connection i guess..
You need to contact Cloudflare support and ask them to fix their faulty ssl certificate..
I will do the same too and ask them to update the certificate.. :-)