Processes in Microsoft 365 for setting up Office apps, redeeming product keys, and activating licenses.
Hello,
You want to remove the old university account from macOS
Select Finder > Applications > Utilities > Keychain Access.
Right-click the account that you want to remove.
Select Delete [account name].
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Then please use the instruction here: https://support.microsoft.com/office/how-to-remove-office-license-files-on-a-mac-b032c0f6-a431-4dad-83a9-6b727c03b193 Download and run the license removal tool to clear the cached organization credential, this should then allow you to sign in using your personal Microsoft account.
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If for any reason it still doesn't work, I need to know your macOS version, because it can be a separate issue that's also going on at the same time.