The address I'm going to delete is a POP account.

First of all, you can only backup messages which are still on the server. When you've used POP before, this is not necessarily the case. The default with POP is to remove messages from the server.

In addition with POP it's only possible to access Inbox on the server. In order to also backup mail on the server in other folders you'd need to set up your account as IMAP. With IMAP you can access all folders and download your messages to the 'Local Folders' account built into Thunderbird.

Once all messages have been downloaded to 'Local Folders' you can delete both, the account on the server as well as in Thunderbird.

Don't forget to create a backup of your Thunderbird profile folder after downloading all messages. https://support.mozilla.org/kb/profiles-where-thunderbird-stores-user-data#w_backing-up-a-profile

I can change my soon-to-be-deleted account to IMAP. Once I do that, I'll follow the notes above and download everything. Thanks for the help!

POP will have all downloaded and sent emails on you computer.

What other emails have you got on your server that you can only see if you logon to webmail account via a browser. Those emails will obviously not be in Inbox. Perhaps you have sent a few emails using webmail or stored some in a webmail folder.

Whilst there is nothing wrong with setting up IMAP, I would not bother with setting up IMAP for an account you are about to delete/stop using. Logon to webmail account and move those emails in other folders into the webmail 'Inbox'. Then in Thunderbird use 'Get Messages' to download them using POP mail account. Then move to appropriate folders.

Then at a later time, you move everything into eg: Local Folders OR simply leave them as they are and just stop the POP mail account from looking for new mail. Make sure it is no longer the default mail account.