Every time I log into a site, I'm asked by Firefox whether I want it to remember the password, but it never takes "yes" as an answer.
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Start Firefox in Safe Mode to check if one of the extensions (Firefox/Tools > Add-ons > Extensions) or if hardware acceleration is causing the problem (switch to the DEFAULT theme: Firefox/Tools > Add-ons > Appearance).
It is possible that there is a problem with the key3.db and signons.sqlite files that store the encrypted names and passwords in Firefox.
Rename the key3.db and signons.sqlite files in the Firefox profile folder.
You can add .old to the file names (key3.db.old and signons.sqlite.old) or move them to another folder to make it possible to undo the action.
You need to set a new Master Password after renaming or deleting key3.db and all currently saved passwords are lost.
If that has worked then you can remove the renamed files that are no longer needed.
I don't think that will work, because the problem is when Firefox asks me whether I want to remember the password for the site I'm at, I press yes, and Firefox doesn't accept the answer. Being that I'm already, at that moment, clicking REMEMBER PASSWORD, I don't see how I can also save the screenshot.
Did you try to rename or move the signons.sqlite and key3.db files?
Do you see any passwords in the Password Manager?
What do you mean by Firefox not accepting the password? What error message do you get? How do you know Firefox is not accepting it?
Try to disable hardware acceleration in Firefox.
Create a new profile as a test to check if your current profile is causing the problems.
See "Creating a profile":
If the new profile works then you can transfer some files from an existing profile to the new profile, but be careful not to copy corrupted files.
@feer56: I mean that the option to save the password comes up and I click yes, and nothing happens. The box I clicked in to indicate that I do want to save the pw remains open and the question is still there. Then sure enough, the next time I go to the same site, I'm offered a chance to save the pw again, because it wasn't saved the prior time, even though I (tried to, anyway) clicked yes.