Did you previously use Firefox 19, and if so, did it have this same issue?

The fact that two browsers are affected suggests there could be an external agent at work. Programs such as CCleaner maybe clear some browser settings. Can you think of any security or system maintenance software that could be at work here?

When you start Firefox, does it have a purple Firefox button, or a purple "mask" icon at the right end of the menu bar? If so, you are starting up in private browsing mode, so neither cookies nor history will be remembered. You can switch that off by changing your preferences here:

orange/purple Firefox button (or Tools menu) > Options > Privacy

Change from "Never remember history" to "Use custom settings" and uncheck the box for "Always use private browsing mode"

I'm not exactly sure when Firefox 19 ended and 20 began. I think it was within the last week. I may have been having problems with 19, but I don't think so.

The Firefox button is orange and I already checked to make sure I was not using "private browsing".

I tried changing the "expiration date" of the cookies to "when Firefox closes", but that didn't help. I added Exceptions/Allow to 3 of the websites I'm having problems with, but this too seems to be ignored.

I have Microsoft Security Essentials handling security issues; I've never had any problems with it.

I think Microsoft Security Essentials is okay, doesn't concern itself with Firefox cookies at all.

It is possible for your cookie database file to be corrupted, but that wouldn't explain why the cookies are cleared so aggressively even within the same session. Still, it might be worth a try at renaming it and letting Firefox build a new one.

Open your current Firefox settings (AKA Firefox profile) folder using

Help > Troubleshooting Information > "Show Folder" button

Switch back to Firefox and Exit

Pause while Firefox finishes its cleanup, then rename cookies.sqlite to something like cookies.old.

Restart Firefox and see whether cookie storage is more permanent.

You might also double-check the feature to clear private data on shutdown to make sure cookies aren't included.

orange Firefox button (or Tools menu) > Options > Privacy

If you have "Use custom settings for history" there is a checkbox to turn this feature on and off. I recommend OFF, but if you are using it to clear other data, click the Settings button to the right to make sure you aren't clearing cookies.

The cookies.sqlite renaming didn't help; no change after file was recreated. I am not clearing private data on shutdown.

I'm out of ideas. As a test of how Firefox runs "uncustomized," could you create a new Firefox profile? In addition to bypassing current add-ons and custom settings, it should have completely fresh settings databases and a fresh cache folder.

Close Firefox and start up in the Profile Manager as described in Profile Manager - Create, remove or switch Firefox profiles. Any time you want to switch profiles, close Firefox and return to this dialog. I recommend using the default location suggested, and not deleting anything without making backups first.

Do cookies work any better in the new profile?

Thanks for your efforts. I'll try creating a new profile, but this one is only a day or 2 old. I uninstalled/reinstalled Firefox 2 days ago.

How would I go about going back to Firefox 18 or 19? Do you think this is worth it?

I would distinguish between your profile (personal settings folder) and the Firefox program files. Reinstalling the program files doesn't clear your settings (e.g., bookmarks). You are given the choice to delete them when uninstalling but I don't recommend that; the new profile test is much less disruptive.

You can get Firefox 19.0.2 from the link in the following support article if necessary: Install an older version of Firefox ("I still want to downgrade"). You should be able to install over your current installation without uninstalling first. Of course, a backup of your settings folder isn't a bad idea: Back up and restore information in Firefox profiles.

Which security software (firewall, anti-virus) do you have?

Try to boot the computer in Windows Safe mode with network support (press F8 on the boot screen) as a test to see if that helps.

I'm using Microsoft Security Essentials for anti-virus and firewall. I tried booting in Safe Mode and there was no change.

I think the problem may be something with my computer and not with Firefox. My computer sometimes gets hung-up when logging off. I've had to cycle the power to get it working again on several occasions in the last week or so. I don't know my way around computers so I'm not sure what I can do to fix my problem. I'm open for any suggestions.

In any event I want to thank everyone for trying to help.