What you need to do is create a new, attached account for the new callsign.
Log into your existing account and go to the 'My Accounts' page,
https://www.eqsl.cc/qslcard/MyAccounts.cfm .
At the bottom of the page you will see a section called: 'Register a New Attached Account'.
Change the callsign to be your new one, put in the start date of when you received the new callsign, and press the 'Create New' button.