I have this too. I can get around it by manually creating a new FaceTime but clicking on an existing one or on a favorite contact won’t let me proceed. I’ve already tried turning FaceTime on and off and resetting. Next 2 steps before giving up and contacting support (probably take a long time), are logging out of iCloud and then maybe resetting the phone entirely. I’m unwilling to do those things so far but next time I get an iPhone I plan to set it up from scratch instead of restore from a backup.
this is not the only hard to troubleshoot issue I have with my phone, but ultimately I’m going to live with this paper cut error since I can get around it. I hate these forums because the replies are usually snarkier than helpful. “Contact Apple support” has a time and place but it’s very costly in terms of time!