I was frustrated at first by your reply: since when does an email client use cookies? Well, dog my cats, turns out that Thunderbird does have some cookie settings that apply to web access: who knew? When I set it to accept cookies, it let me give my Outlook password and authenticated me. So everything's cool now.
Suggestion: this document that was linked in the email I got from Microsoft about needing to update my authentication method doesn't say anything about cookies. (It does have a section pertaining specifically to Thunderbird, but only covers changing the authentication method.) Can someone contact the team responsible for that document and see if they can't add something there about enabling cookies? I'm sure I'm not the only person who will have been bitten in the butt by this problem, with no idea how to fix it. (The setting option in Thunderbird is at Privacy & Security--> Web Content--> Accept cookies from sites.)