Since you'd already tried that, Harleyhans, we have a couple of more steps to use.
How to use safe mode on your Mac - Apple Support -- Restart your iMac in safe mode and see if you're able to sign into FaceTime. Safe mode does require more time to start up, and the first paragraph in this article explains why. After testing, restart your iMac normally.
New user or group setup - Apple Support -- If you still can't sign in using safe mode, create a new Admin user account and see if you're able to sign in.
Get back to us when you're able to with the results of both tests please.
Take care.