Using the New Outlook for Mac for personal email and calendar
Thank you for the update.
Since you can receive emails normally and can send emails through Shaw Webmail, the issue appears to be with Outlook rather than your Shaw email account.
Because you recently changed your password, Outlook may still be using an old credential for the outgoing mail server.
I would try removing the account from Outlook again and then adding it back using the current password. If that does not help, please check the outgoing (SMTP) server settings in Outlook and make sure they match Shaw's current settings.
When you try to send an email, do you receive any more detailed error message or error code than "Connection to the server failed or was dropped"?
Also, are you using the latest version of Outlook for Mac available through Microsoft AutoUpdate?