[Edit] To answer my question, yes, Dropbox does indeed store its application information in your user’s /home folder by default. As long as you don’t wipe your /home folder, you should be good to go once you reinstall the Dropbox app after reformatting/reinstalling your distro (Tested with a few Fedora-based distros, YMMV if you use Debian/Arch). I didn’t have to re-login; the Dropbox app just worked.
Is it possible to reinstall Linux (or distro hop) without losing my Dropbox install? Could I move the Dropbox install to my home folder so it survives the OS install?
Well, that makes a huge difference to the meaning of the question.
I don’t know, but maybe the login is held in a dotfile such as ~/.dropbox or maybe in ~/.config/dropbox or similar, and just backing up that (not to Dropbox!) would be enough to restore being logged in on a different system.
New problem: they have 2FA as well
If my suggestion works, that won’t matter, it will still be logged in on the new install.
Fix: download what you need locally, reinstall Linux, find a different software or account, be free of this problem forever.