My mozzarella always morphs into the shape of the container I've got it in...with a plastic bag, there is no structure to help it hold any kind of shape.
Ice water to firm it up first is always good. And it can take longer than you think because the interior of the cheese is going to be quite warm from the stretching. That warmth will move outward, so even if the exterior feels cold after a few minutes in the ice water, there's still heat to come out. When it is warm, it will change shape.
Mozz is one of the most challenging cheeses to make, so give yourself a pat on the back for being successful with your 2nd batch!