I had a similar issue with underclocking and my HD 7850. Had artifacts popping up all over my screen while on the desktop and the card was idle.
What eventually fixed it for me was to use a CCC profile to set my minimum core clock to 500MHz. No more artifacts.
Create a new profile in CCC and check the boxes at the bottom, save but do not activate the profile. Find the XML file it creates. For me it was in C:\Documents and Settings\(name)\Local Settings\Application Data\AMD\ACE\Profiles (or something like that, doing this from memory at work - and yes, I still use XP). Open the file in Notepad and change the following line from:
<Property name="Want_0" value="30000" />
<Property name="Want_0" value="50000" />
Save the file, go back in to CCC, and activate the profile. I used GPU-Z to monitor the change in clock speed and to keep an eye on the temperature. You may need to experiment with the minimum core clock speed between 400-500MHz for stability. 500 was my magic number.
Hope that helps.