Sorry to hear you're still having problems with this. <.< By the sounds of it I'm guessing you're playing the v0.14.5 download? Since the newer versions of the game have episode/chapters instead.
Just trying to confirm which version and all the tech details to look into it further, because v0.14.5 was our last version of the game using Tyranobuilder as our engine. While anything above that uses Ren'py engine and has been much more stable in comparison, smoother, etc.
I've tried to download v0.14.5 and run some tests to recreate this issue. The only thing that I have found that might be the problem and worth mentioning. When you downloaded the game, you are exporting the zip file/folder "Messy Academy v0.14.5" folder from the zip folder itself onto your desktop or into another folder on your computer, right? If you are trying to run the game app inside the zip folder itself, it can't change the contents of the zip folder to keep saved data inside of it. (I only ask this and suggest this because I tried doing this for testing and it was the only thing that led me to the similar issue you're having.)
I've attached a screenshot to show you what the game folder should look like after you have exported it from the zip folder onto your desktop as a example. (This shows the extra folder/files after I have already ran the game once and saved, creating saved data which I highlighted to point out.)
Hopefully this helps! ^^ 