I feel your pain and have a solution that should work. I have been battling this for several weeks now, finally gave up, and wrapped up a call with an Apple engineer an hour or so ago. Come to find out, regardless of what the O/S tells you in Activity Monitor about Safari, Dashboard, or both "not" running, these processes are in fact being called. It seems that, at least in my case, on an up-to-date OS X / Safari combo, and with The Weather Channel widget installed, you will never get the newest Adobe Flash Player update to install. As soon as you remove The Weather Channel widget, the Adobe Flash update process will again work normally. It makes absolutly no sense, but I and the Apple engineer were able to reproduce the fix on my two systems and completely reproduce the issue and the fix on his. Not certain where the "blame" for the issue rests, but I suspect this is a issue with The Weather Channel widget - or at least how it is interacting with the O/S and Safari. I suggest checking for any installed/active widgets that grab/stream any soft of content via the web, removing them, and re-running the Flash updater. I hope this helps. Please reply to let me know if this did the trick for you too.
Thanks, tmr3075. I'm sure this is the fix, as all the time in the past when Flash would update just fine I had Chrome set as my default browser, but now I have Safari as the default (probably another good reason to go back to Chrome). I finally managed to kill the Dashboard Client and that let me install the update, but as you say, probably Safari was the actual culprit, since I also had the Weather Channel widget running in my Dashboard (and that uses the default browser). But when I was using Chrome, Flash didn't care since Chrome didn't use Flash. If I decide to keep Safari as my default browser, I'll kill the Weather Channel widget. Thanks so much for solving this!