It would be interesting to know whether Firefox's password manager works with those two sites for other users, but I suspect the odds of finding one here are low. Maybe those sites have a discussion section where the issue has been or could be raised.

If you still have interest this is what I found from one of the past episodes for this issue. The problem is FF no longer gives access to Cookies Exceptions in Privacy. This did solve the problem then. The complete history of that issue can be found here. https://support.mozilla.org/en-US/questions/990921?page=1

'''Details like websites remembering you (log you in automatically) are stored in a cookie.

So you now appear to keep the cookies from those websites.

This should have nothing to do with the places.sqlite file.

Create a cookie 'allow' exception to keep such cookies, especially in case of secure websites and when cookies expire when Firefox is closed.

   Tools > Options > Privacy > Cookies: Exceptions 

In case you are using "Clear history when Firefox closes":

   do not clear the Cookies
   do not clear the Site Preferences 

   Tools > Options > Privacy > Firefox will: "Use custom settings for history": [X] "Clear history when Firefox closes" > Settings
   https://support.mozilla.org/kb/Clear+Recent+History 

Note that clearing "Site Preferences" clears all exceptions for cookies, images, pop-up windows, software installation, passwords, and other website specific data.

Clearing cookies will remove all specified (selected) cookies including cookies with an allow exception that you would like to keep. '''

If you use the default option of [Remember History] you do not see the option because there is nothing to be an exception. If you click the dropdown arrow to use Custom Options then you get further options including the option to set exceptions.

If you use some third party software to clean up or clear cookies that could still mess things up.

Look at this article that includes screenshots

• Enhanced Tracking Protection in Firefox for desktop_how-do-i-change-cookie-settings

I have found a way to keep auto sign in to work partially on the 2 sites in question.

In options, privacy, history, use custom settings for history, always use private mode, accept cookies from site, exceptions.

i have copied the urls for the 2 sites and pasted them into exceptions. This retains the auto fill in function as long as the computer is turned on. When shut down and rebooted i lose all these setting and the History reverts to Never Remember History. Not sure why I can't keep the settings.

There does not seam to be a way to SET the settings and keep them after a shutdown.