I would add the earthlink email account to Thunderbird, using IMAP, and then start copying messages to either a working account or to Local Folders.

Email is not well suited to moving huge blocks of data. I'd suggest you do this folder by folder and let each one complete before moving on to the next. I'd manually create each folder as needed. Copying folders (particularly by drag-and-drop) seems to be a bit unreliable; right-click and "copy to" is rather more robust. It can be slow; be patient.

As a footnote:

IMAP will create a mirror replica in Thunderbird of the mail that is on the Earthlink server. They are linked: if you then were to delete the messages from the server, that deletion will be synchronized to Thunderbird. Boom. Mail gone.

If you want to make a backup copy of the messages that would survive a server deletion, there will be an extra step of copying the messages into local folders (which are independent of your synchronized folders).