I ended up opening a second free account with another email. While I can't have Evernote open twice, I can have the old Evernote open in a browser while I am using the new account in the Evernote app.
I've just started entering new, better-formatted notes into the new account. If I need to grab something from my older notes, I can copy it from the browser (images don't copy that way though). If I really needed to, I assume I could log into the old account with the app, export notes, log out and and then import them in the new one.
Someone's also told me that Evernote search is considerably better with a paid account. Possible, but my notes were so badly structured that search was useless, so I strongly considered abandoning Evernote - hardly a motivation to try to pay for it first.