bbs.spamgourmet.com

For a while (I think more than a month?) the site will accept my username and password, telling me I'm logged in; but as soon as I try to do anything, the next page that loads says I'm not logged in. I have tried waiting (obviously) and resetting the password, but the issue persists. At this point I need to get in and refill one of my addresses fairly urgently.

I've not seen this happen with SpamGourmet, but with other sites. Try a different browser or different computer or both.

that sounds like your browser isn't accepting session cookies from the site - note, these are different from persistent cookies, which are the ones that everyone should be concerned about. Session cookies disappear when you close your browser. Unless you check the box that says to remember your login - then you get a persistent cookie.