After trying a million other things, I tried this and it worked. Went into Yahoo (Profile>Sign-in info), scrolled to "Secure mail key>Manage secure mail key," I chose the e-mail account I wanted, and selected "Add secure mail key." Once the "secure mail key" was created, I copied it. Then I went into Thunderbird, having saved all my old messages as files and removed the account, and recreated the account, pasting the "secure mail key" as the password. Two weeks of new e-mails immediately downloaded. Then I imported all those old messages, and things seem to be working just right.

Thanks for all your good suggestions leading up to this point.