Your handbrake adjustment is the same as your old car. Just remove the handbrake cover, the bit with the ash tray, and you will see the same two long threaded bolts with two nuts (on each thread) that tighten against each other. You just need two 10mm (i think) spanners.
Logic suggests that if the rear pads are worn then the handbrake will travel further. Yours sounds the other way. Probably worth getting the mechanism and cables checked when you fit the discs and pads, but probably just needs a bit of adjustment.