See second comment below for instructions.
Charr, and anyone else having trouble, I just tested these instructions and they do really work. The secret is being sure that you sign in to your google account BEFORE posting. then you must UNCHECK the box that says STAY SIGNED IN. I just did a test by trying to see if i could sign in and click ON that box, but it doesn't let me post. So if you follow those instructions it does work. I think only pc users have this issue. I don't know why blogger doesn't fix this. It used to not be a problem until the last few months.