The 560 (even though outshined by the 660) is still a very capable card. I'm only using the 560 non-ti but if I clock up to 950/4400 (and I really haven't found any use for it as it runs games fine), I should be about the same as a 560 ti.
If you can consider a 660 non-ti, factory oc'ed, that would be great.
560 vs 660, 660 will win and with less power draw and only requires 1 power connector. Still, the 560, as I would consider, is the base line and is a very good choice for the price to performance ratio at 1920x1080.
I think it would really boil down to the pricing. If it's only the 560 in your price bracket, it's a good choice but if you can reach out a little, the 660 is well worth the price bump.
On average, an overclocked 7850 will be on par with an overclocked 660 (and will easily outperform a 560). However averages don't apply to every game (hence, being averages); some games favor Nvidia cards over AMD cards, and vice-versa. Which will be better for you largely depends on what games you'll play, but either one will likely give you more than satisfactory performance in any game.