First you should email Support@swanbitcoin.com from your new email address. In your email please mention your old email address and that you want to change your Swan email to your new email address.
We will then respond to your support email with a link to prove your identity in a short thirty-second process.
Shortly thereafter, you'll get an email that your account is frozen. Please follow the directions in the email to unfreeze your account, after which your new email will be your primary email on your Swan account.