I have the same problem.
What I found is that quitting and restarting FaceTime app prompts me to log in to FaceTime (as if Keychain forgot the password?). Then the app just "stalls". So I quit and restart FaceTime and it prompts me to confirm my identities (so, one step further). Then the app just "stalls" again. Finally if I quit and restart FaceTime once more, I have my contacts list and can make or receive a call.
I go through this three-starts dance each time I want to use FaceTime and it's a reliable workaround.
Is this the same for you?