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To make it appear as if email is being sent from your alias address, you need to change the 'Reply To' or Return Address (depending on the email software used) to your alias rather than your 'real' email address.
You may wish to set up another personality, or account, (again, depending on which software is used) then you can switch from one to the other.
Do not change any other mail settings. You will continue to send and receive through your ISP. But with the 'Reply To' changed, it will appear as if the message is coming from your alias at your domain name.