Usually when any regulator leaks in this manner, it's the orings around the adjustment portion, or on the regulator piston. Those are usually the only two areas air can get out the bottom(adjustment side) of a regulator. It's usually a pretty easy fix.
To the OP, once you have the reg taken apart, don't be afraid to look around with your manual in hand and learn what the parts do. You're going to have to lube the whole thing at some point so you may as well learn in advance
Learning is fun!