I'm not really sure if you need an other PSU, but 350 doesn't sound that great. I think you should have at least 500W. I heard it's best to have a single +12v rail with a stable ampere output. I have an coolermaster GX 750W with a single +12v rail with 60A max and i'm pretty happy with it, but i'm no expert on this. I paid something like 80 euro for it so that's well under 200 pounds. More expensive PSU are a bit better in power efficency and are often modulair, wich means you can disconnect the cables you don't use so they don't interfere with the airstream in your pc. ( i think that's only importent in really hard pushing game pc's wich get pretty hot )
Otherwise ; just put in the 6870 and see what it does, i think it also depends on the rest of your system and what kind of games you play.