Adapters in general should not hurt as long as you have enough power(those 2 work the same, just one takes from SATA while the other takes from LP4), what else is in the system or did you get the recommended bundle as well(cpu ect).
Do you have Vsync on or off? What screen resolution? What settings? ect. A 5770 should be ok with that game.
My guess is that the PSU isn't providing enough power to your video card and I bet that adapter isn't helping things either. Just buy a PSU with 2x 6-pin PCI Express connector and enjoy your gaming. Depending on if you want to upgrade to crossfire or your entire system in the future I would consider a a higher quality PSU but if its just for gaming with that video card I think you'll be good with the $40 range.
Please post your entire system specs so we may help you better.
Components seem good. These are the possible scenarios I can think of:
You have low space on your SSD or HDD and you're using too much RAM, you have page file enabled and since you have low space on your SSD or HDD things will perform shitty, but Windows should give you a warning when this happens so I'm guessing this isn't it.
PSU isn't giving your card enough juice on the 12v rail.
Funny thing here is it says 25 amps @ 12 volts, but the listed wattage under it is only 237(19.75 amps)
25 amps would be 300 watts. What do you trust now.
That is what they call BS in the industry.
I would also like to point out that if it only has 19.5 amps, it is worse then my 300 watt power supply(22 amps). I know for a fact that runs a 5770 without issues.
I am not 100% sure here, but things done look too good for that power supply(despite all games I have played on a 95 watt cpu + 5770 being under 200 watts at the wall, but I am not sure I can even trust a power supply with such messed up ratings), I would be more interested in game settings as well. Vsync on vs off(source engine does some strange things) for one as well as turning down or off any AA.