Please tell me you've at least set up the email forwarding and "send-as" features so that you can send/receive via that account from another account if you ever want to? And on that note, maybe also set up an application-specific password if it allows you to do this without entering a password? (Though MAKE SURE to test this on your new account to make sure it doesn't lock you out with a password prompt if you attempt this.)

(Although note that if you do this via Gmail it will leak your sending email address in the headers.)

Also, I think you might be able to retrieve the rest of your emails too. You might not have IMAP access, but at worst, you can write Google Scripts and have then upload your emails to another account. It will be rather slow but you should be able to copy all your emails this way. I don't think it should prompt you for a password when giving permission but I'm not sure; again, test this on your new account before trying. Make sure to wait like half an hour to make sure any internal password prompt timers expire.