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There are quite a few things to look at and consider if you want to play games. First is the higher the resolution the worse your experience is going to be, especially in your budget range. Next is going from a 15.5" to a 17"+ is going to cost at least $100 more.
Next major item is that if you want the best experience for gaming you will want a system that has a Discrete graphics card and not one that is built into the CPU.
With that said I will be giving you a few solid suggestions that in the 15"-16" range. I will be using Newegg since most laptops ship for free or for less than $10.

First item, This is $699, designed for gaming, no tax and has $3.99 shipping. http://www.newegg.com/Product/Product.aspx?Item=N82E16834152586

Second item, an older model with less slower CPU but with faster GPU and not designed for gaming but will have better gaming performance. It will have a more plastic-y feel. $629 with $3.99 ship and no tax.
http://www.newegg.com/Product/Product.aspx?Item=N82E16834314535


Both of these machines will perform much better than your existing one and will be better options for gaming than the ones you had listed.

I personally would go with the Acer, it is the best bang for your buck that I could find throughout all of Newegg. It may be early the early 2014 model but it should last. It has a 15.6" screen running 1366x768 resolution. You are going down in screen size and resolution but the picture will be just as clear due to the size difference in the screen. With the minor compromise in resolution you should be able to play pretty much all modern games on Medium or High settings and get fully playable FPS. I hope this helps.