This used to happen to me all the time with this American Lock 700 series I got. I'd have to use a light tension wrench to pick it and a heavier one to turn it.
Let me ask you this, where are you putting tension? Top of keyway or bottom? As in, spot closes to the pins or farthest away? If you're putting tension on the bottom of the keyway (farthest away), try going to top of keyway. It'll be weird at first, but it gives you better torque on the plug since you're tensioning it in a more centered spot.
This is all assuming you already know the lock IS picked and you're just having trouble turning the plug.

Good luck.
Rob