If you use "Delete cookies and site data when Firefox is closed" and not "Clear history when Firefox closes" then you should keep cookies with an allow exception.

If that isn't working then check for issues caused by extensions.

Start Firefox in Safe Mode to check if one of the extensions ("3-bar" menu button or Tools -> Add-ons -> Extensions) or if hardware acceleration or userChrome.css is causing the problem.

• switch to the DEFAULT theme: "3-bar" menu button or Tools -> Add-ons -> Themes
• do NOT click the "Refresh Firefox" button on the Safe Mode start window

• https://support.mozilla.org/en-US/kb/troubleshoot-firefox-issues-using-safe-mode
• https://support.mozilla.org/en-US/kb/troubleshoot-extensions-themes-to-fix-problems

A detail like a website remembering you (log you in automatically) is stored in a cookie.

You can use these steps to make a website recognize and remember you:

• create a cookie allow exception with the proper protocol (https:// or http://) to make a website remember you
• use "Delete cookies and site data when Firefox is closed" to let cookies without allow exception expire when Firefox is closed
• Options/Preferences -> Privacy & Security
  Cookies and Site Data: "Manage Permissions"

In case you use "Clear history when Firefox closes" or otherwise clear history.

• do not clear the Cookies
• do not clear the Site Preferences

• Options/Preferences -> Privacy & Security
  Firefox will: "Use custom settings for history":
  [X] "Clear history when Firefox closes" -> Settings
• https://support.mozilla.org/en-US/kb/delete-browsing-search-download-history-firefox

• clearing "Site Preferences" clears exceptions for cookies, images, pop-up windows, and software installation and exception for password and other website specific data
• clearing "Cookies" will remove all selected cookies including cookies with an allow exception you may want to keep

Three notes:

(1) While you are on the site, if you click the lock icon, does the site information panel show a custom cookie permission has been set?

(2) The Storage Inspector tool (usually Shift+F9) will list the cookies set by the current page. You can take a look at the list to see whether your exception is working: the Expires column will have a date in the future and not the word Session. (There are some cookies that a site may set intentionally for the session only rather than Firefox shortening expiration to Session. An exception doesn't help with those.)

(3) It may be easier to create a cookie permission using the Page Info dialog. This doesn't always work because sometimes cookies are set for a base domain instead of a www domain, but it's a convenient shortcut for many sites.

While you are on the relevant page, you can call that up using either:

• Ctrl+i (for Mac, Command+i)
• right-click a blank area of the page > View Page Info
• (menu bar) Tools > Page Info

When the dialog comes up, click the Permission icon at the top to show that panel.

Scroll down to "Set Cookies" and uncheck the "Use default" box, and then select Allow to let the site set persistent cookies.

There's no Save button, so you can close the dialog after making the change.