If you are oversteering on entry, try less rear droop. Also look at your radio settings and try to turn down your steering expo. What radio do you have? I think on Futaba and Airtronics/Sanwa radios you will use negative expo. Maybe around -30/40%. On the Spektrum DX3R it is positive Exponential. This will slow down your steering when you first turn the wheel so that it will not be as twitchy. Also check your front droop as well. This will affect steering when getting on the throttle. You can also try some camber gain in the rear. Do this by adding shims under the rear hub. Or take out shims from the inner camber link on the rear. From my experience, this mostly affects mid corner and exit rear traction.
Brushless, I use SpeedPassion or Express Brushless.
Oh, and I do not have any problems with the battery moving side to side. It is down tight. And pushed all the way in up against the servo mount and motor center brace. So it doe not move at all.