Can you show screenshot of what you see, when you cannot sign in?
Have you tried to use "enter serial number" instead of signing in?
If AutoCAD license information on your Mac was corrupted for some reason, you may also try to clean uninstall (if you have not tried this way):
https://knowledge.autodesk.com/support/autocad-for-mac/learn-explore/caas/sfdcarticles/sfdcarticles/...
before installing AutoCAD again.