Can't get rid of VERY old xbox account

First of all:

Go to
https://account.live.com/Proofs/Manage and change the password of your account.

Then make sure you turn on Two Step Verification and set up the free Authenticator App.

With Two Step Verification and Authenticator App, a hacker would have to physically access your phone to get a secret code from the app to get into your account from then on.

After you secured your account that way, you can go to https://account.xbox.com/Settings/Home/ and then click on Xbox 360 Profile Protection, and click the button to require
a profile download for the Xbox 360 consoles shown.

This will remotely disable your Xbox Profile on the Xbox 360 console you gave away.

And if it's on a Xbox One, changing your password and setting up Two Step Verification is enough to disable your profile on the Xbox One you gave away.

That can happen only if you close your MS account permanently: https://support.microsoft.com/en-my/help/12412/microsoft-account-how-to-close-account

Make sure to personally clear all important and personal information.

Change the password and add an Alias to sign in.

Don't forget the 2-Step Verification Factor.

Yikes hackers are never fun. If you are still able to access the account on PC you should be able to mark it for closure through the security settings menu.

All you need to do is change the password.

Have a look
here to protect your account even if you don't use it or have access to your console.