That will be because your plasma didn't do 'black'. The GZ950 does and represents coming out of black as it should. I.e. dimly.
In what lighting conditions are you watching your GZ?
You don't want to change Brightness at all as it's perfect at default. You'll not gain anything, in the near black response (1% to 10%), by increasing Contrast but, you may want to increase Luminance a smidge, instead of decreasing it. This may not improve near black response but, I find OLED's typically require a higher peak output to look similar to the other screen technologies, with lower outputs.
You might be out of luck as the TV is already performing as it should, where shadow detail/ near black response is concerned. The tools anyone has to improve this are lacking too. The lowest level that can be adjusted, in the White Balance and Gamma 'Advanced' menus is 2.5IRE (should say %) so, you're still missing the adjustment for the darker areas. the 2 point white balance controls may provide the adjustment required though.
10 and 20% are a little high to adjust, if you're trying to boost near black response. You should look at the 2.5 and 5% controls instead. And adjust these in the Gamma menu, for your purposes, as the White Balance controls are for just that, balancing the white levels.
Paul