Been playing with the Dualshock 4 I bought when they were released oddly a week before the system. I've had to do a couple repairs on it, but have yet to have L2 do anything besides select the left half of my hotbar. I would try recalibrating; I know you said you tried resetting the controller, but if you're on PC, did you reset on FFXIVs client, Windows, or both?