I think I'm worried about the relaxed swing because my instinct tells me that the ball will fall if I don't cup my wrist.
My biggest question is if there is a way to squeeze the ball anywhere to keep it from slipping on a relaxed swing. Do you apply pressure with your thumb, fingers, both, or none?
If the ball is slipping from your hand you need to add tape to the thumb hole. You want to grip like you would to keep a bird from flying out of your hand if you had it's legs between your fingers and thumb. Meaning you use the pads so the nails don't dig in and hurt the bird and you don't squeeze so hard that you break it's legs.
If you put so much tape in the hole that you have to screw your thumb in the hole and it still feels like it is going to slip off then you need to start going forward toward the palm with your thumb pitch. Take the ball back to your driller and have him/her put a slug in the hole (if possible might have to plug), with 1/4" more forward thumb pitch than you are currently using, ie. if you have 3/8" reverse try 1/8 and so forth.
Before moving your thumb forward you need to make sure that your span is not stretched, some drillers like to stretch spans for what ever reason, most can't tell me why just that its how they've been doing it for blah blah years.
You can tell if your span is correct by placing your thumb in the ball and laying your fingers across the holes. The leading edge (side closest to the thumb) of the finger hole should be no longer than half way between the first and second knuckle of the middle and ring finger without stretching your hand, it should be relaxed. It can be a little shorter than that, but not shorter than one quarter from the second knuckle.
If you already have a ball you can tell if your span is right by putting your hand in the ball and see if the fingernails are pressing the back of the holes, a properly fit ball will not have the fingers pressing into the back of the hole.
On my equipment, I can put my thumb in first and still be able to put my fingers in the ball, I donít advocate bowling like this, but it is a good test to see if your span is too long.