Are you supposed to be able to save in between sessions? It works if I quit to the menu and then hit "continue", but if I quit the app entirely I lose all my progress. (I'm running Windows 10, if that helps)
There are three possible issues here.
First - the saves are bound to the game file. So if you moved the swf file to another folder, or changed it's name - the saves will reset. Quick fix to reverse this is just to move the file back to the original folder or change the file name to as it was when you first played the game.
If the above is not the case - there's the second issue - in order for the game to save - you need to have local storage enabled in your flash player. Remember, this is still a flash game that requires flash player to function properly. Quick fix is to right click on the game screen, then click on settings and enable local storage option.
Read more about local storage and learn how to fix it on this flash player help page.
If that also doesn't resolve your issue, here goes the third option: certain antivirus software can block the file, since it's an .exe (or .app) and allow it to run as an untrusted software. That can cause the savefiles not to be stored. Make sure the game you bought is on a white list of your antivirus.