Actually, you're incorrect. Logging into your Avast account using the old email address, then adding your NEW email address to that account as the primary should correct this issue, as long as you remember the password for the Avast account associated with the old email address.
In order to make your new email address the default for the account where your subscription is active, you HAVE to login to Avast with the old email address first, then add your new one. This will send a confirmation email to the new email address. After confirmation, you can tell your account to make the new one the default, and then remove the old one from your Avast account. From this point on, you should be able to login to the correct account, using your new email address. Once you do this, just login to avast using the new email, and you're done
NOTE: you may have to close the avast account that is currently using your new email address before adding it to your old avast account. Worst case, temporarily recreate the old gmail address, long enough to retrieve the password for your old avast account