In case anyone is interested, I've figured out how to make this work.
There are three files to edit (I suggest making backups of these files):
order.php
tpl_checkout_shipping default
tpl_checkout_confirmation_default.php
In order.php change the following:
$sql_data_array = array('customers_id' => $_SESSION['customer_id'],
'customers_name' => $this->customer['firstname'] . ' ' . $this->customer['lastname'],
'customers_company' => $this->customer['company'],
'customers_street_address' => $this->customer['street_address'],
'customers_suburb' => $this->customer['suburb'],
'customers_city' => $this->customer['city'],
'customers_postcode' => $this->customer['postcode'],
'customers_state' => $this->customer['state'],
'customers_country' => $this->customer['country']['title'],
'customers_telephone' => $this->customer['telephone'],
'customers_email_address' => $this->customer['email_address'],
'customers_address_format_id' => $this->customer['format_id'],
'delivery_name' => $this->delivery['firstname'] . ' ' . $this->delivery['lastname'],
'delivery_company' => $this->delivery['company'],
'delivery_street_address' => $this->delivery['street_address'],
'delivery_suburb' => $this->delivery['suburb'],
'delivery_city' => $this->delivery['city'],
'delivery_postcode' => $this->delivery['postcode'],
'delivery_state' => $this->delivery['state'],
'delivery_country' => $this->delivery['country']['title'],
'delivery_address_format_id' => $this->delivery['format_id'],
to this:
$sql_data_array = array('customers_id' => $_SESSION['customer_id'],
'customers_name' => $this->customer['firstname'] . ' ' . $this->customer['lastname'],
'customers_company' => $this->customer['company'],
'customers_street_address' => $this->customer['street_address'],
'customers_suburb' => $this->customer['suburb'],
'customers_city' => $this->customer['city'],
'customers_postcode' => $this->customer['postcode'],
'customers_state' => $this->customer['state'],
'customers_country' => $this->customer['country']['title'],
'customers_telephone' => $this->customer['telephone'],
'customers_email_address' => $this->customer['email_address'],
'customers_address_format_id' => $this->customer['format_id'],
'delivery_name' => $this->billing['firstname'] . ' ' . $this->billing['lastname'],
'delivery_company' => $this->billing['company'],
'delivery_street_address' => $this->billing['street_address'],
'delivery_suburb' => $this->billing['suburb'],
'delivery_city' => $this->billing['city'],
'delivery_postcode' => $this->billing['postcode'],
'delivery_state' => $this->billing['state'],
'delivery_country' => $this->billing['country']['title'],
'delivery_address_format_id' => $this->billing['format_id'],
In tpl_checkout_shipping_default.php remove or hide:
<div class="buttonRow forward"><?php echo '<a href="' . $editShippingButtonLink . '">' . zen_image_button(BUTTON_IMAGE_CHANGE_ADDRESS, BUTTON_CHANGE_ADDRESS_ALT) . '</a>'; ?></div>
In tpl_checkout_confirmation_default.php change:
<div class="buttonRow forward"><?php echo '<a href="' . $editShippingButtonLink . '">' . zen_image_button(BUTTON_IMAGE_EDIT_SMALL, BUTTON_EDIT_SMALL_ALT) . '</a>'; ?></div>
<address><?php echo zen_address_format($order->delivery['format_id'], $order->delivery, 1, ' ', '<br />'); ?></address>
to this:
<address><?php echo zen_address_format($order->billing['format_id'], $order->billing, 1, ' ', '<br />'); ?></address>
Hopefully someone will find that info useful.