Going by parts:
Windows 7 system which came out to only a strange 5.9
Windows 7 benchmark base the results on the lowest result only, probably your HD is 5.9 and the rest of the system get that score also... nothing to worry about. that is very common.
Then I also ran AIDA 64 which reported a core clock speed of 300 Mhz, instead of 1 Ghz that I think is the standard for the 7770.
You must check the clocks when the card is in use, run a benchmark program and while running it check the clocks again.
Not feeling happy, I then went for the 3D Mark basic trial and here are the results...
Have you already upgraded your graphics card drivers to the latest version? that changes a lot the graphical score.