You didn't say what other hardware is also getting power from the 350W PSU.
It's well below the 450W PSU recommended by Sapphire for it's 4770 cards.
Even more of a concern is the 9A +12V rail. Most 'average' 350 PSUs will have 19A or better +12V rating.
I've seen $10 budget 250W PSUs with more than 9A.
If that 9A is accurate and you're powering a normal PC set up I'd say you probably can't run the 4770.