I'm running a 6970 and the 5970 on a Corsair TX850. I thought the red light could also mean overheating because it lit up when the card overheated.
What motherboard? That PSU should be plenty beefy enough, and has a single rail, so unless you've melted your connectors somehow it should supply plenty of power. Make sure connections are snug of course.
If you're using an older mobo it could be a problem supplying the 75W PCI-e slot power..
Does the card behave this way if you take the 6970 out?