Aaaaaaakkkkkkk!!! Can you believe I lost MY response to this thread? Not because of the cookie problem, but because of the link thing you brought up? I had half a dozen windows open and closed the one with my reply because I thought I had already posted it.
Anyhow, the drift of it is that what you described fits the cookie problem to a "T" and now that you deleted the cookie and got a fresh one written hopefully it won't recur. But if it does, I need to know and the circumstances, because that means it's a problem that's not gone yet.
Regarding the logout button, you might want to read this thread:
Where's the Logout Button?I added it now to the recommended reading list:
Which Topics You Should Actually readHopefully the board's working smoothly for you now.

ETA - BTW, the "Delete cookies" link on the footer menu doesn't really delete the cookies. It just cancels the current session ID so the logged status switches to logged-out. To solve the cookie problem, you need to actually to DELETE the cookies using whatever tool is available to you. I'm assuming you know how to do that, but if not, let me know which browser you're using and I'll give you a step-by-step.