Offline/local saves (so on your machine) get stored in so-called flash cookies, more technically known as Local Shared Objects, or LSO's. They're different than cookies proper, but get treated by your system more-or-less the same way.
If no savegame gets stored for you (you did click "continue" there, right, not "new game"?), you're not properly allowing cookies to be set for ArmorGames. Check your browser settings, as well as your FlashPlayer settings. FlashPlayer needs to be set to allow third-party content to be stored, as well.
As for cleaning your cache, you will similarly need to set it so that it doesn't clean up the cookies in question. The same goes for any other cleaning tools you use, whether automated or by hand.