Sending, receiving, and organizing email in Outlook.com
To better understand what's happening, could you please clarify a few details? Are you accessing your email through Outlook.com in a web browser, the Outlook desktop application, or another email application such as Windows Mail or Apple Mail?
When you're asked to try another server, what is the exact error message? If possible, please share a screenshot. Does the password prompt appear inside your email application, in your web browser, or as a Windows security notification?
Please open https://outlook.live.com/mail/ directly in your browser and try composing a new email addressed to yourself. If you can send the message successfully from the website, this would suggest that the problem is related to the email application's configuration or authentication rather than your Microsoft account's ability to send messages.
Also, signing in using your fingerprint does not necessarily mean your Microsoft account password is being accepted, since fingerprint authentication may use Windows Hello or a passkey.
Please share your findings to determine the appropriate next step without making unnecessary changes to your account.