OK, this is what I'd do (plenty of experience with WEIRD ATI/AMD driver issues!

):
1 - Download Display Driver Uninstaller:
http://www.guru3d.com/files-details/display-driver-uninstaller-download.html
2 - Download latest drivers for your card from link John provided.
3 - Disconnect from the internet (remove Ethernet cable or disable internet adapter temporarily).
4 -
*Important* Once both are downloaded and you're offline, reboot PC into Safe Mode by hitting F8 when Windows is booting and selecting Safe Mode when prompted.
5 - Run Display Driver Uninstaller - select 'Remove AMD/NVIDIA Folder' and make sure the option to reboot is selected. You don't need to boot back into Safe Mode. This will proceed to completely remove your display drivers and all associated software.
6 - Install your new drivers once you have rebooted. The reason I've told you to disconnect from the internet so that Windows Update doesn't try and download 'updated' drivers for you.
7 - Reboot after installing drivers. Hopefully you are now good to go. Connect back to the internet.
This is the correct method to completely remove and reinstall graphics drivers if you are having problems like you are.
It's worked for me on several occasions when my AMD Catalyst drivers have gotten really screwed up.
I don't think your card is failing by the way. I think you are just having some driver problems which should hopefully be fixed by simply correctly removing the old ones and installing the new ones.
One question - you said before you had two of these cards. Are you running them in CrossFire or are you just running one of them? If you are running them in CrossFire I'd recommend taking one out and seeing if the problem persists.
Post back and let us know how this goes.