The card is only a 6800 , 350 is fine for that. (foot in mouth here , althought that also depends on how mmuch you are runing , i.e dvd/cd drive , anything else that draws power) Although it would not hurt to upgrade the PSU if you got extra dough , may as well go for a 500w.
It does sound like various game issues , face it nearly every game is relased with fingers crossed that it is going to work on every individual system out there.
I have spent hours on forums in the past trying to get various games to work. You can try messing with the settings inside your games graphics options, especially look for anything that mentions shading or shadows , make sure to turn those to
low or
off. I have a high end vid card and have to turn shadows to thier lowest to avoid problems. It's often not the as much the hardware involved as it is the game itself. An example of this is that HL2 works fine while some of your others do not (Thank You Sierra and Valve for making such a kick ass game !)
You can also try looking for hombrewed shader patches if there are any for your game/s.
Remember the key thing about video cards is not the amount of ram they have , it is the GPU and what shaders they support that really matters.
A good example is NWN2 , a graphically beautiful game but Obsidian just had to go out and use some exotic ass shader that only certain cards deal with nicely , even other high end cards have had problems runing this game.
Blame the game companies I guess. Some folks will tell you to buy an Xbox or whatever but it's not nearly as fun , or frustrating
You could also search for Black Widows guide to tweaking your PC for gaming. He will tell you all of the BS services you can turn off , just little things to free up resources , it does help your games run smooth.