Hi Alan,
Since you still know the passwords for both Microsoft accounts, the challenge appears to be the verification process rather than the account credentials themselves. In situations like this, it can help to check whether either account is still signed in on another trusted device or application, such as Outlook, the Mail app, Microsoft Edge, Xbox, or a mobile device.
If you can find an active session for either account, you may be able to access the account's security settings or receive verification requests there, which could help break the circular recovery situation. On the affected PC, you can also review Settings > Accounts > Email & accounts to see which Microsoft accounts are currently connected to Windows and related services.
If neither account is accessible through an existing signed-in session, there is unfortunately no supported way to bypass Microsoft account verification requirements, even when the passwords are known. In that case, the official account recovery process remains the recommended path for regaining access.
I understand how frustrating this situation can be. If you can confirm whether either account is still signed in on another device or application, we'd be happy to suggest the most appropriate next steps based on that scenario. Best of luck, and we look forward to your update.
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