Looking like this is a tough nut to crack.
The static could be caused by so many things, as it is an onboard device it may be more sensitive to cross talk from other components, it could also be a faulty component.
Sadly I cannot think of any other things to look at, you have tried drivers, checked your settings, proved it is not TS itself at fault, and confirmed all audio cables (including the CD/DVD drive audio cable if fitted) are secure and not routed past high EMF emittance devices such as the GPU..
As refitting the sound card appears to be a non-starter it looks as though you may have to explore the possibility of using a USB head set