Can I send WhatsApp messages only when an order ships, and not when it is placed?

Yes. Each order event has its own switch on Plugins → WhatsApp Notification, under Order Notifications:

1. Untick Order Placed, Order Confirmed, Order Delivered and Order Cancelled.
2. Keep Order Shipped ticked and enter its Template name exactly as approved in Meta.
3. Add its Variables (CSV of order fields) if the template has placeholders.
4. Click Save settings.

Make sure Enable WhatsApp notifications is still on, or nothing is sent at all.