Well, , I missed that one. You can edit /includes/modules/pages/checkout_one/jquery.checkout_one.js, finding
Code:
    function changeShippingFields(event)
    {
        jQuery(this).addClass('opc-changed');
        jQuery('#checkoutOneShipto .opc-buttons, #opc-ship-save').show();
        jQuery('#checkoutPayment > .opc-overlay').addClass('active');
        jQuery('#checkoutOneShipto').removeClass('visibleField');
        jQuery('#checkoutOneShipto').addClass('opc-view');
    }
and changing to
Code:
 
    function changeShippingFields(event)
    {
        jQuery(this).addClass('opc-changed');
        jQuery('#checkoutOneShipto .opc-buttons, #opc-ship-save, #opc-add-ship, #opc-add-ship+label').show();
        jQuery('#checkoutPayment > .opc-overlay').addClass('active');
        jQuery('#checkoutOneShipto').removeClass('visibleField');
        jQuery('#checkoutOneShipto').addClass('opc-view');
    }
If your site is using minified scripts, you'll need to push that unminified version through a minifier; I use https://jscompress.com/.