I would say it doesn't sound like the PSU. If the card was getting power through the mobo only the card should sound a warning beep to let you know it's not getting power through the other two plugs. And as you say the fan is spinning so it's getting power from somewhere.
TBH it sounds like the weight of the card might be pulling it away from a few of the pins in the PCI-Ex slot. I've know this to happen before. Try taking the card out, laying the machine on it's side and reinstalling the card, making sure you get it straight. In fact, while the card is out, check the slot for dust and/or bent connections.
Hope this halps buddy