Can't play .swf files in Firefox after upgrading Shockwave Flash Plugin...

There's a couple things that might be happening.

After getting a ton of feedback about requiring people to exit Firefox as part of the installation, we changed the process to let you update with Firefox running; however, this leaves open the possibility that you have open tabs with the old plugin, and new tabs with new plugins.

You *really* want to exit Firefox completely and restart it after an update.

The other historic reason for double-plugins is that something either got corrupted in the Windows Registry, or in Firefox's internal registry.  It's typically the Firefox config file that gets weird, but let's eliminate Flash from the equation first.

You can manually go in and blow away any remaining Flash artifacts by following this guide:

https://forums.adobe.com/thread/928315

If you find that you can't delete one of these keys, it's because the registry entry is corrupted.  Windows runs it's registry checker automatically, so you'll probably need to turn to a commercial third-party registry cleaner/repair tool if that's what you're running into.

Before you do that, you probably want to take the opportunity to make a good copy of backups, then do some basic disk maintenance.

Check your hard disk for errors - Windows 7:

https://support.microsoft.com/en-us/kb/2641432

If you've completed the "clean" uninstall, there's really nothing left on the machine that comes from Flash.

I'd just reboot at this point to allow the system to run the registry checker and clear out anything weird that might be in memory (this really eliminates a ton of potential things that could go wrong),

Then download and reinstall Flash from here:

http://get.adobe.com/flashplayer

If you're still seeing erroneous entries in the Firefox plug-in list, it's time to either move to a clean Firefox profile, or just do a clean Firefox install.  You can export all of your information and import it back to Firefox, but that's beyond the scope of what I'm going to write here.

Here are a couple options that you can check out:

Use the Profile Manager to create and remove Firefox profiles | Firefox Help

Refresh Firefox - reset add-ons and settings | Firefox Help

Also, just as a piece of general advice, you should really just install the latest Flash Player and opt-in to Automatic Updates.  We're constantly doing proactive security work, and running an old version increases your risk of malware infection.  Automatic updates will ensure that you have the latest fixes as soon as they become available, which is particularly important when we're pushing fixes to problems that have been discovered in the wild.  Oh, and if you've disabled Protected Mode, you should really turn that back on.  It's important.