Ah, I feel really,
really stupid right now! I've done everything from registry fixes, uninstalling and reinstalling a billion times, and even reverting back to Flash 9, all to no avail. I even checked my sound card and drivers. I finally managed to get it fixed just a few minutes ago, and when I saw what was wrong, well... face, meet palm.
So anyway, what I did to fix things was very simple: I went to Control Panel > Sound and Audio Devices. In the audio tab, I noticed that the default device under Sound playback was
not my sound card. I don't know how or why it changed, and I could have
sworn I had already checked that setting. Anyway, I changed it to my sound card and saved the settings and voila! Flash now has sound. Two weeks of frustration and that was all it took to fix it. >_<
I still don't know why it only affected Adobe Flash, though, since all files played in Mplayer worked fine. I guess it's a mystery...