The USPS 1583 form does not initiate mail forwarding.
The 1583 form authorizes us to receive mail on your behalf. In order to have your mail forwarded, you will need to complete a Change of Address form with USPS. https://moversguide.usps.com/
If you do not wish to have all of your mail forwarded to us, you will need to manually update your mailing address with anyone who sends you mail. You can do this easily by using the "My Address" tool in your account (see video below). This is ideal if you want only certain types of mail to go to your Earth Class Mail address.