Your experience is unusual - and poorly described. It is therefore something specific on your system, which can be fixed.
I had similar issues on prior macOS/iOS systems and typically were solved by logging out of iCloud and logging in back. Somehow, the certificates used by the system to authorize connections became invalid for some tools. This can be provoked by system upgrade when more than usual number of things change, but does not need to be.
Try logging out of iCloud - you may need to logout both in system Preferences (which credentials are used for Contacts, which is what Messages are using) as well as Messages and likely FaceTime. You may want to uncheck and recheck various options you have selected there to refresh settings, in case they go corrupted. If it still cannot be sorted out, it may be configuration (plist with configuration) for Messages is corrupted or Keychain is corrupted. There is instructions on like (google it) how to reset those.
In any case, it should be working. Most people have no problems.