There are two Only Once! plug-ins: an authentication plug-in and a user plug-in.
Before uninstalling both these Only Once! plug-ins, you must first enable the corresponding Joomla! plug-ins, that is: both the Joomla! authentication plug-in and the Joomla! user plug-in. After you have disabled both Only Once! plug-ins, they are no longer active and your website behaviour should return to 'normal', but ONLY if you have enabled the standard Joomla! authentication and user plug-ins.
For the rest, it is not clear to me what exactly the problem is that your users are experiencing upon exiting. If they are not let back in by Only Once!, they must have another session open, but not from the same work station. But again, although I would love to solve the problem, it is not clear to me what the exact problem is....
By the way, the fact that you have the freeware edition is not the cause of the problem.