Does a Radeon HD 5830 really need 2 6 pin PCI power cords?

The GPU has rated power maximum power of 175W.
16x PCI-E slot ca provide 75W.
6-pin PCI-E auxiliary connector can provide 75W.
So yes, this card requires two 6-pin PCI-E auxiliary connectors.

If your PSU is rated at 34A or higher on the combined +12V rails, you should be fine to use molex to PCI-E power adapters for the extra connector if your power supply doesn't have it.