I haven't see that error with XP on my home machine yet. But there's always time.
From what I gather, the only browser that handles flash correctly is Chrome. So updating it from Adobe might not be the answer for flash issues to begin with.
For those who don't know, Firefox has a built-in function that examines installed plugins for whether they're in need of updating. Find it under Tools / Add-Ons / Plugins section. Immediately above the list of plugins is a link "Check to see if your plugins are up to date" which opens an information screen with links to the source vendor if an update is available.
Firefox extensions and appearance add-ons can be checked similarly for updates but via a picklist menu on the gear icon.