Hi,
onShippingSave is called when the shipping method selection is changed on the legacy checkout but that isn't the case anymore with the new checkout system.
I suppose that you were previously using the old checkout system and the 4.4.0 removing the "checkout legacy" system lead to moving to the new checkout and the mechanism you were using before doesn't work properly anymore because the saving is handled differently on the new checkout system.
The goal of that function was to check that the selected shipping method was indeed available to the customer.
However, this check is now handled directly when saving the checkout without the need to call a function of the shipping plugins as the cart system already knows what are the available shipping methods.
What you're saying seems to indicate that you were using onShippingSave for something else.
For example, maybe you were adding your own HTML to the shipping method selector with custom_html and in that case, you have the trigger onShippingCustomSave(&$cart, &$shipping, $group, $data) you can implement but this supposes that you HTML has input fields with names like checkout[shipping][custom][$group][$shipping_id][name]