I just bought an ASUS 7850 DCU2, and installed it on my old computer (AMD Phenom II X3 720 with 4th core unlocked, running on stock speed, 2x2GB DDR2, Gigabyte MB, Acbel 470W+100W PSU).
But all games got same problem: screen flickering or part of it not shown at all, suddenly show other part/area, stuttering..
when I disabled the 4th core everything going fine again.. (before installing 7850, my setting have no problem with on board graphic card and other older graphic card..)..
Looks like a CPU or PSU problem to me. A discrete card will draw way more power than a IGP, putting a bigger strain on the PSU.
From what I understand, what may be happening is that the increased load influences the voltages supplied to the CPU. Not necessarily outside of specifications, but since you unlocked your CPU, it may just be what's causing this.
Either disable the 4th core, or get a better PSU. Increasing voltage on your CPU manually in the BIOS may also help.
More information about the PSU would be good, too.