When I try to delete my iCloud account on my iPhone it tells me to sign out using my old email and password. I don't remember the old password How do I get my iCloud account to let me change my email account without having to give my old password for my other email account? I don't remember that password.
You can easily reset your password if you have access to the email account that you've used while creating your Apple ID. Go to this page: https://iforgot.apple.com/password/verify/appleid and the email your primary or rescue email address and hit the 'Forget Apple ID' button. 

You can also reset your Apple ID password by answering the security questions. You might find this Apple Knowledgebase article useful: https://support.apple.com/en-in/HT201487