In your situation, to do something, there are 2 mandatory requirements. Your old email address must be already verified (at the time when it has been active still). For an account without verified currently set email address, the same can't be changed to another email address. The other requirement is to be able to login your account (somehow). Non authorized person can't access the account, what about change something in the settings (of course, would be strange and insecure otherwise).
To log in your Dropbox account, you don't need active email address. If you can recall your currently set email (as seems to be, doesn't matter if inaccessible) and the password (and other authentication data, if any), you can login - no problem. If you have forgotten your password, but still have a linked mobile device to your account, usage of the feature "Connect a computer" in the mobile device settings could grant you access. Here "computer" is not mandatory be a real computer; you can use any device (including mobile) with installed supported web browser. If there is no way to gain access, ... there is no way!
If you can't change your email, for some reason, the only way would be create new account using desired email and move all your data there. Take in mind that the new email, you want to set to the existing account, must not be assigned Dropbox account yet; otherwise gonna fail!
We love to learn from the educators who use Dropbox. Whether you teach kids, teens, adults or a combination of all three, we want to know what apps and integrations you use with Dropbox to help with teaching. Which of the ones below is your favorite, or most used tool?