Hello,
You do need to add a new alias, make it primary before you can remove the current alias.
Can you sign into your personal account via this link: https://account.live.com?
If so, go to Your Info > Sign in preference, Add an alias.
You have two options here:
Create new: this will allow you to use the native domain outlook.com to create a new alias.
Add existing: this will allow you to add a 3rd party email address, such as yahoo, gmail, as long as they are not already associated with another Microsoft account.
Once added, make this new alias your primary alias, then you can delete the other email as alias, and that email is free from this Microsoft account instantly.
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Please tell me why it doesn't let you if you don't mind to be a bit more specific. What happened after you clicked on Add an alias, how far did you get to, what error did it give you?