You would usually get steering shake with warped discs, does it do it if you pump the pump the pedal a few times before braking? if it doesnt then this shows there is air in the system
"Also when applying the handbrake once stopped, when I let go of the foot peddle the car seems to shift slightly till the brakes catch. "
this sounds normal because the footbrake is holding the front and rear brakes, so as as you release it it will move slightly as the suspension moves, if i'm reading what you wrote properly