I guess a 6850 would be a noticeable improvement, though not a earth shattering one.
Meeting the 200 bucks or less is kind of hard as most the performance series cards are 300+
I would probably go with this one Newegg.com - XFX HD-685X-ZNFC Radeon HD 6850 1GB 256-bit DDR5 PCI Express 2.1 x16 HDCP Ready CrossFireX Support Video Card with Eyefinity
It comes with dirt 3 as well.
The biggest difference between the two cards is the 6 series handles dx11 much better.
Though even a 5850 would be a noticeable step up from the 5770, it's just not really a gamer's card.
Actually come to think of it, you can get the 6870 for basically the same money right now so might as well spend the extra 20 bucks plus this one has a 30 dollar rebate actually making it the cheaper of the two. http://www.newegg.com/Product/Produc...-506-_-Product
and it has more stream processors which means less choking when shaders or light maps get intense.
It's still not the enthusiast level card, but it's more than adequate to play anything on the market right now.