Rolled back to 2.3e, and it's working fine again. Then, I updated again, and this time, it was fine at first. But after clearing the HFS cookies, the error came back. It looks like it's creating different cookies based on the version. There's a cookie named 'HFS_SID' generated by 2.3e and one named 'HFS_SID_' by 2.3f. As long as the first cookie is there, it's fine, but it writes the HFS_SID_ cookie and then refuses to locate it.
So then I decided to reset the template, and that appears to have solved the issue with recognizing the new cookie name (HFS_SID_). I'm guessing changes made to the template for 2.3f didn't carry over when I upgraded for some reason, and I don't remember editing it, but maybe I did. The clean install would also have solved it (and I was going to do that next).