After seven months of limping along with an ever-so-slightly-broken authentication system, I ripped the whole thing out and replaced it with the official solution sponsored the Phoenix developers.
Whenever it is merged and deployed (which I expect is probably in the next few days), everyone will be logged out, because the old session database is being removed. If you would like, you can also clear your cookies for the site when logged out, but this isn’t required.
You will then be able to proceed to sign in again as normal. The only change is that now, if you click the box to have your session be remembered, you shouldn’t end up being frequently and infuriatingly logged out anymore.