and my current Apple ID is connected to an old, school email that no longer exists,
Indeed, if you no at school anymore, they already deleted your school email address. This is done as soon as you have left the school so that you can no longer log in. You no longer have any rights there.
An deleted email address, it is no longer valid, it simply no longer exists and it no longer works.
You may not be able to use / change that school email address because it is no longer exists.
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You have an Apple ID now.
You are sign in here . . . with your Apple ID & Password.
Your Apple ID is the personal account you use with all Apple products and services. It includes the email address and password you use to sign in, as well as the contact, payment, and security details that you use with Apple.
You can’t merge multiple Apple ID accounts.
What happens when you use the same Apple ID & Password that you use to sign in here?
See -> Sign in with your Apple ID -> Sign in with your Apple ID - Apple Support