Tabbed browsing is now a feature in the latest Internet Explorer, so that's no longer a selling point.
I'm using Firefox with no problems here. It always shows me logged in on the front page. I'll assume you're using the latest version. Try hitting Ctrl-Shift-Del--this will clear all your private data. This won't clear your cookies by default though--you can access all that stuff from the Tools->Options Dialog. Look on the History tab.
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