Hi Amos,
According to your description, a possible cause is that Lync certificate may become corrupted after the update.
Please try the steps below to see if the issue persists:
1. Open the Keychain Access certificate management utility. To do this, in Finder, click Applications, click Utilities, and then click Keychain Access. Or, search for Keychain Access by using Spotlight.
2. In the left pane, click login, and then click Certificates.
3. In the right pane, find your certificate for Lync sign-in, select it, and then delete it.
Note You may have to unlock your keychain by using your password.
4. Close Keychain Access.
5. Restart Lync for Mac.
Also you can try changing the password, here is the steps to do it in OWA:
1. Click the little gear icon on the top of the menu bar in OWA.
2. Click Office 365 settings.
3. Navigate to password.
4. Change your password and save.
For more detailed information about troubleshooting Lync for Mac issues, please refer to this article: Troubleshoot Lync for Mac issues in Lync Online for Office 365.
Thanks,
Henry Huang