I'm having a similar issue. Facetime rings but doesn't connect, and then completely fails after that. I've tried verifying my settings but the settings aren't being saved or I get an error that the connection to the Facetime server is lost.
My turned out to be that the FaceTime servers were down. However I wasn't able to receive incoming calls after they came back online. I had to log out of my Apple ID on FaceTime and iMessage and log back in. I did this for all my iOS devices and my Mac. Everything works fine now.