Hi Robert,
The account used in the Core:Sendmail setting is used to authenticate against the SMTP-server (in your case Exchange) when sending e-mails from the otrs-system.
For Exchange, this means the account you enter needs to have rights on your Exchange server to send e-mails on behalf of all the email addresses in the otrs-system. This is a configuration in Exchange ("send on behalf of"). You can either use your product1-mailbox for this, or create a new exchange-account with rights on all three mailboxes.
Another option is to enable a dedicated receive connector in Exchange that accepts all incoming SMTP-traffic from your OTRS-server. In which case you can configure the connector to bypass the permissions-check entirely.
Hope this helps,
Mark Wesbeek