I am having the same problem using OS X Mavericks 10.9.2 and trying to install Flash 13. I have tried what scant tips I have been able to find:
-Ran uninstaller and made sure there are no Flash remnants around
-Rebooted to my Admin account in Safe Mode
I have read the "How to install a clean Flash install" page and followed those instructions.
The end result: Nada. I tried reinstalling Flash 12 using the offline download and now I get "this installation failed" on that one.
So now I have no Flash on my computer and no way to get it there. I've wasted way too much time on this and am ready to tear my hair out.
I had the same issue, but after a few tries (and restarts), it just started working again. I really haven't the slightest idea of how I got it to work, but maybe persistence helped!
Maybe try restarting after uninstalling everything? I think I did that between uninstalling and trying to install the new version.