7850 would be like 2x better framerates. If you can get it, go for it.
Over the 560? That seems a little inflated. Looks like they're neck-and neck on BF3. Skyrim appears to run better on the 7850, but it's not 2x better, and you still get playable framerates with the 560. ME3 isn't going to tax either of those cards.
So, for the games that the OP wants to play, the 560 is a perfectly sound choice. That said, of course you should get something better if it fits in your budget, and the 7850 is a better card overall than the 560.