I have the mahogany top version, but the guts are the same. After I read your post, I plugged in and ran a finger exercise on each of the five positions- also let it sit untouched with the volume up and ran the toggle through all five clicks... nothing. The guitar was pretty silent.
I suspect your cord, actually. In my experience some pickup types tend to get noisier with even very small breaks in the shielding. My old Ibanez was horrible for it. Maybe pick up a new, good quality cord and try again before sending your T5z back.
FWIW