The same thing is happening to me since we went over to the EA App.
I've tried uninstalling and reinstalling it about six times now, and I've tried running it on Steam instead.
But the same thing keeps happening:
1) I open a New Game, make a Sim and move them into a lot. Maybe I build a little house. I don't do anything major; it's just a test run.
2) I save the game, give the save file names in various formats (so it's nothing to do with what I'm calling the file), and carry on for a bit.
3) I try to save the game again before I quit, and get a message along the lines of "could not save your game (Error Code 16)."
4) I have to quit without saving. I close TS3 entirely and go back to my desktop.
5) I open up TS3 again and try to open the save file I made.
5a) If the file's there at all, it won't open. The interactive loading screen comes up, but flashes past in a second. I get the "serious error loading <save file name>. It is recommended that you restart the application" message.
5b) Sometimes the file isn't there at all.
6) I try to open another New Game, and the interactive loading screen flashes past in a second. I get the "serious error loading <world>. It is recommended that you restart the application" message. This happens no matter what world I try to open.
I've tried to contact EA Support but they just suggest clearing the cache, or uninstalling the game and reinstalling it. Neither has worked.
TS3 is my favourite game, too, and I just can't get it to work. I've put hours into trying to fix it at this point, and nothing is taking.
I wish I could help you, OP, but all I can tell you is that you're not alone. I'm just not sure it's something to do with the EA App, because it's been happening via Steam as well.
One thing that strikes me might be a problem is that I sync my Documents folder to Google Drive, and that's where the "Electronic Arts" folder is. (But I don't have similar problems with TS4.)
Additional context: I'm on an HP Pavilion Gaming Laptop 15-dk1xxx and I use Windows 10. TS3 is installed on the 476GB C-drive, which has 173GB free.
So, it doesn't seem to be a lack of storage space, which is usually the cause of Error Code 16, apparently, and it's not like I'm using an ancient laptop with a tiny hard drive. I bought this computer specifically to run TS3 with high-end graphics, and it always worked until December 2022, which is when I found this problem. It's completely wrecked the game for me.