Also of note is that the shadow layer is not showing up at all, which kind of ruins the entire game.
Someone had good luck fixing this with a similar card by updating their graphics driver. Here's what you need for
Windows 7-64 bit. Here's what you need for
Windows 8-64 bit.
Inexplicably, they are different driver versions and different versions of the Catalyst Control Center for the different OSes. These drivers work for me on an even older ATI card (3870), so perhaps this will work for you? Let me know the results!
Also, if you are running a different OS, I can try to find the appropriate driver for that.