Hm. I've tried deleting all local browser storage sources that I can but primal.net seems to hang on to the session. I'm using the Alby extension, so maybe that's what's doing it. Perhaps pasting your nsec gives you the option to "log out" in the Primal UI whereas with an extension you are expected to use the extension UI to do so.

in that case, it would be Alby extension > "slider" icons on the Primal tile > scroll down to "Disconnect"