I don't know about always but it's said that for awhile. Browsers have been removing Flash functionality. Can you not download from the Play page? Does the player I linked to not work? If you don't see your saves, you just need to find them and transfer them to the new folder.
On PC it is possible to place a .cfg file in
C:\Windows\SysWOW64\Macromed\Flash
and
C:\Windows\System32\Macromed\Flash
to get Internet Explorer to play SWFs from specified websites.
I don't know if it is possible to do something similar on Mac. It probably depends on if your browser still supports Flash at all. What browser are you using?
If you want to try, save this file as a .cfg and try to find the corresponding folder(s).