This is really a new issue to us. We are investigating and will try to figure out the problem soon. The different browsers shouldn't be making a difference, but it's possible. When you click the logout button, it should (as it says) log you out of both your site and Facebook and kill your cookies for both in the process.
If you determine anything new, let us know. Also, if you would, try on our demo site and let us know if it occurs there. If there's something we need to change in our configuration to make it more like yours, let us know so we can narrow it down.
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