catsmound Does it help to put the EA App in offline mode first, and disable any notifications and game mode-type settings within the App? This shouldn't matter, but maybe something about how the App works, graphically at least, is conflicting with the game.
If that doesn't help, please remove Options.ini, in Documents > Electronic Arts > The Sims 3, and try again to play. if you see the same graphics issues, exit to the Main Menu and change any settings you want to there, then quit to desktop before loading a save. Be sure to disable Advanced Rendering and enable Object Hiding while you're there.
If that doesn't help either, please post the full specs of your Mac: everything under About This Mac including the OS and except for the serial number.