Whenever I save the game it says that it's been saved, but when I go to the save game or load game screen I see nothing there. I have also tried to turn the save automatically option on and off, but that doesn't help either.

Where have you got the game installed? If in "Program Files" or "Program Files (x86)" try uninstalling and then reinstalling somewhere like C:\Games to see if that helps. Also make sure you are running the game as administrator. Hope this helps.

Well, I don't have a C:\Games folder. Where is it located?

You create it yourself.
Click on your C: drive in Explorer and right click. Select New > Folder and then rename the new folder, Games.
If you are on Windows 10, your problem may be that your Documents folder has been redirected to Onedrive.
Your Documents folder has to be on a fixed Drive (not cloud) for DAO to see them.
If you used the GOG Galaxy client, then your problem, like so many others could be caused by Galaxy.
In that case I would reinstall the game with the Classic (non Galaxy) installers and in C:\Games as Pajama said.

How do I install it using the classic installers that you're talking about? Sorry I am new to GOG.

I'm not exactly sure because mine defaults to the classic installers, but it you click on DAO on your account page it should say More and then classic installers.