First, I'd make sure I have the latest drivers from Nvidia, but it sounds like you have done that already. I never trust the drivers from the OEM card maker.
Second, there is the possibility of a damaged card. I don't know how your card was packed when shipped to you, bubble wrapped but otherwise loose in the box or molded plastic, keeping the card from moving. I just got done buying a 6800GS. I got an eVGA 6800GS wrapped only in bubble wrap. It was broken (capacitor on the end of the card broken off). I sent it back. They sent me another. It was similarly broken. I sent it back. They then refunded my money. I ordered an XFX 6800GS instead this time. It was in a molded plastic container and was fine upon arrival and works great.
I never plugged in the two broken cards. They might have worked except for odd problems for all I know. So, one suggestion is to make sure there are no capacitors either broken off or having connections snapped at the end of your card. The capacitors are the silver cylindrical things near the end of the card. Here's a picture that shows the capacitors. The vulnerable ones in shipping are the silver cylinder-looking items with purple half-moons on the top.
http://www.newegg.com/Product/Showimage.asp?Mode=&Type=&Image=14-130-265-02.jpg%2C14-130-265-03.jpg%2C14-130-265-04.jpg%2C14-130-265-05.jpg%2C14-130-265-06.jpg%2C14-130-265-07.jpg&CurImage=14-130-265-03.jpg&Description=eVGA+256-P2-N391-AX+Geforce+6800GS+256MB+GDDR3+PCI+Express+x16+Video+Card+-+RetailAnother issue might be that the card is overclocked too much. If you did the overclocking, perhaps you turned it up too much. Either way (whether you or the manufacturer did it), you can use Rivatuner to turn down the clocking if you want to play around with it. Rivatuner is free and available here. I wouldn't play with the clocking, though, unless you can't return the card and try another, as this might void the warranty.
http://www.guru3d.com/rivatuner/Also, with Rivatuner, you can play with fan speed and see if the high-pitched whining you hear correlates to fan speed. If so, then the fan is your issue. For whatever reason then, your sound card or audio wires are picking up electromagnetic noise from the fan. You could try moving your audio wires or making sure your headphones are plugged into the card at the back of your PC directly, not into the front jack on your computer.
Another possibility perhaps, if the card is very new, is to send it back and get another of the same brand or try a different brand of 6800GS (such as XFX).