I could tell you a build that runs all games on high detail and can get 180 fps (but I cap it to 60). It costs £300, unsure about Dollars, if you want to know just say, but it uses AMD parts and I know what you mean about trouble with them, I'm currently building a Intel only PC with Nvidea graphics :S
For £300 I don't think you'd be able to get a build anything close to being able to run new demanding games on full settings even near 30 frames..
As for the OP GPU choice, the 6850 is a great card (i had a 6870 which is just a bit more powerful) and so is the 560 ti. The 560 ti is more powerful and will last longer and will overall be able to get pretty noticeable framerate increase from the 6850 but if you're tight on money then go with the 6850. Both are great cards, money and what brand you want comes in to play more though.
Im afraid you can Take my best friends PC for example, £326 for him to buy the parts, it runs Battlefield 3 and Skyrim at 120 frames. Ill ask him for a list of his parts when I see him next. He is a very.. PC nerdy person >.<