I am currently running 1920 x 1080 with dual eyefinity. I would definitely consider upgrading the power supply.
This is what you must understand; any graphics card that money can buy that will run on your current stock Power Supply is simply a waste of money. You will definitely have to upgrade your Power Supply if your planning on getting a new graphics card.
Since your running on 1920 x 1080; I would recommend the following;
Power Supply: Corsair Enthusiast Series 650W V2 [Cheaper than the Corsair 550W]
Price: $89.99 + Free Shipping / Before $20 Rebatehttp://www.newegg.com/Product/Product.aspx?Item=N82E16817139020
Graphics Card: XFX 6870 1GB
[Great price/performance ratio, handles most modern games at high/medium settings @ 1080P]
Reasons: This card costs $10 more than the other 6870s after rebate, I felt the dual fans would be much more efficient at cooling the card during heavily loaded operations. Even though the 6870 is last generation, it offers much more performance for the price compared to the current lineup. And being older comes with much more drivers, a win win if you ask me.
Price: $189.99 + Free Shipping / Before $30 Rebatehttp://www.newegg.com/Product/Product.aspx?Item=N82E16814150521
Here is a 5750 VS 6870 benchmark. We know for a FACT that the 5750 is faster than the 6570. See for yourself the difference it makes;
6570 vs 5750;http://www.gpureview.com/show_cards.php?card1=653&card2=616
5750 vs 6870;http://www.anandtech.com/bench/Product/540?vs=537
Edit: While $200 may seem like a lot for upgrading, but you'll gain much need performance to boost your gaming experience. The 6870 will be able run most modern games at great playable settings.