Hi Anita, that's strange. Could you cross-check the History setting? On the same part of the Settings, page, scroll down to the History section.

If the selector doesn't already say "Firefox will: Use custom settings for history", go ahead and choose that to make all the checkboxes available.

The last row should have "Clear history when Firefox closes" and a Settings button. If this box is not checked, then Firefox is not clearing cookies when it closes. However, if the box is checked, click the Settings button to see what categories are listed. Does it work any better to un-check Cookies and Offline Website Data here?

Firefox does allow users to create an option user.js settings file to override your settings changes every time Firefox starts up. The following article has that and some other relevant things to check:

How to fix preferences that won't save -- "Specific preferences are not saving" section

Check privacy.sanitize.sanitizeOnShutdown and privacy.sanitize.pending in about:config, make sure they are not locked and do not exist in a user.js file in your profile folder.