@Erica_S1 this is not what the person means.
You used to be able to start a screen recording with the microphone enabled, but the OP i discovering that doing this tried and true method that may websites teach you how to do is not working. Audio is being not being recorded while there is a FaceTime call in progress.
This used to be possible, but it seems that this "feature" has caught the eyes of Apple execs and both the latest macOS and iOS are now preventing it. Yes, Live Photo seems to be the only option left right now, but this is not the answer the OP was looking for. The answer is that this ability has been removed.
The only solution is to use an older device with an older macOS or iOS version that still supports this.
This is a real hassle because there are many legitimate reasons to do this, and in many locations legal too.
Another solution is to record the audio on another device and then merge the audio and video together in something like iMovie.