Some days the Plugin Check page - https://www.mozilla.org/en-US/plugincheck/ - is overloaded with traffic and is very slow to respond by providing the links for where to get the recommended updates from.

You should probably just by-pass waiting for the button for Flash to appear and just go here instead via a bookmark for this page. https://www.adobe.com/products/flashplayer/distribution3.html

Flash has had 5 security updates in the past 2 months (three in January alone), which is very unusual even for Flash - one of the most used and most exploitable Plugins out there . Last year I think they went close to 9 months without needing to do any security updates at all. Quite honestly, Adobe may need to tweak the built-in Flash updater set-up to look for updates for their program more frequently than the once a month interval they now have set. Or come up with a better, and more up to date update mechanism - which hasn't ever worked for me on WInXP for as long as I have been using WinXP.

there appears to be a bug with the plugin-check page as well where the loading won't lead to a result. in order to resolve that, go to the firefox menu ≡ > addons > plugins & set the "java deployment toolkit" to never actovate...

Thanks for the help! My initial problem that I had already updated via the link that the-edmeister posted and Flash was still not working quickly resolved itself. As for the plugin thinking circle never stopping, philipp's solution fixed that immediately! I have no idea why, but there you go!