You must have an old or very crappy 450 Watt PSU, it's more like a 300 Watt unit at best. 22 Amps on the +12V rail is not likely going to be enough for your 6770 and the rest of your system. If you use a second PSU to help power your graphics card, that might just barely be adequate if you put your system under load, but you will be cutting it very close, your graphics card will still pull most of its power from the motherboard and your "450 Watt" power supply. The secondary power supply will only give your card the wattage needed above the 75 Watts pulled form the slot, which would be about 33 Watts for a 6770.
I strongly recommend that you get a new power supply before you try to use your 6770. Pushing old or crappy PSUs to their absolute limits is not a good idea, there is a risk that you could damage your components if one of your PSUs blows. Look for a good quality 400 or 450 Watt unit. Something in that category should have roughly 30Amps between its +12V rails. That will adequately power your system.