Here's how to do it:
- Go to Dropbox and log in with your current email.
- Click your profile icon (bottom-left) and select Settings.
- Under the General tab, click Edit next to Personal email.
- Click Send email to set password.
- Go to your email and click Reset password.
- Set your password by typing your new password, then submit.
- Now go back to Dropbox then go to Settings.
- Under the General tab, click again Edit next to Personal email.
- Type your new email address and your Dropbox password.
- Click Update email, then confirm change. Your new email will be active on your Dropbox account.