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It sounds like you are automatically being logged into Facebook because your web browser is storing the user name and password information. If you deleted the account from Facebook you wouldn't be able to login to that account anymore. You should be able to logout of Facebook, and login with a different user name. You can also delete the stored user name and password from the web browser by navigating to the privacy settings and clearing the history. Deleting all the old cookies may also help refresh the browser. CCleaner is a free application that'll do most of that work for you accross all installed web browsers on your system.
https://www.piriform.com/ccleaner/download
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