You don't have to log out to post anonymously, actually. You just press reply on whatever thing you were going to reply on, and then "more options." That'll take you to a page with well, more options, more ways to format, and a better looking preview. From there, you can select what account you want to post from if you have multiple, and if you want that post to be anonymous.

I agree with the bit about any fill I write ending up somewhere else which makes it less anonymous, I usually post an AO3 link, so. That ruins the anonymity, I suppose.