I have a model 52, but embarrassingly have never shot it
. My understanding from those who have is that it is very sensitive to "follow through" when shooting it, as mentioned above. The fact that it prints without vertical stringing off a rest is more evidence of this.
I suspect you are on the right track getting stocks that provide a consistent hold; a cheap alternative may be to put adhesive skateboard tape on the front and backstraps, as long as the adhesive doesn't damage the finish. Consistent and repeatable grip pressure with a secure hold will likely fix your problem. Good luck!