Unresponsive iMessages and FaceTime on Mojave
Have you tried deleting files in the ~/Library/ folder? ('~/' means your user folder)
search facetime there, try deleting what you find.
It might sign you out, at the same time it might solve your problem.
You can search 'Messages' there as well.

I don't guarantee it won't do any harm. Yet it's what I usually do when I run into problems. It's sort of DIY mentality.

If you don't see the Library folder, it's because it's hidden by default.
You can change that in "Show View Options" menu in "View" option of Finder. Or by pressing command and j at the same time. (You should be at ~/ for this option to be shown).