I'm using Firefox as well, and am having no issues.
https://support.mozilla.org/en-US/questions/1319622
https://support.mozilla.org/en-US/questions/1097453
I'd try the first one and restart firefox and check if it worked or not, then the second.
You have to be careful if you use firefox's password manager, because clearing the cache/cookies? deletes your saved passwords. Make sure to restart firefox after doing this stuff, just to make sure the stuff takes effect.
Other than that I don't know what the issue could be, I thought login stuff was all related to browser cookies, but there could be some website specific issues I am unaware of.
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