I have a bow that twists on the bottom limb if I pull it back and then un-pull-it-back, but when I shoot it, it lines up just fine. It could be that I'm torquing the handle when I pull it back. And maybe you're torquing the handle somehow.
I would think that when you shoot it, and it finds its own way, that's where the string is truly centered (or not), and you should make adjustments based on that.
So I'd shoot it, and see where the string lines up after the shot, and make small adjustments until it's tracking down the middle.