Jeff - It sounds like you either have a motherboard issue or your graphics card isn't getting good power from the power supply, but it's difficult to be sure. If you go to a good computer shop (the type where they fix stuff), they'll probably let you plug your graphics card into another computer for free or cheap just to be sure, but the first thing I would try is a different power supply just because that's an easy thing to try and it's consistent with your issues.
ExK4's comments