Here's the answer I got from Shopify support. (It doesn't solve the problem of all the outstanding partially paid orders and the solution going forward adds six new steps to every order in question. That's not a solution.)
"Therefore, as a workaround, I would highly suggest for you to create a draft order with the exact shipping cost moving forward:
- From your Shopify admin, go to Orders.
- Click Create order.
- Add products to the order.
- Customize the order: Add a customer, Apply discounts, Turn taxes on or off, Add tags or a note.
- Do one of the following actions:
- Email an invoice to the customer. By default, the invoice contains a link to a checkout that your customer can use to pay for the order.
- Accept payment from the customer.
- Save the order as a draft so that you can update it later from the Drafts page.
Here's our help documents about Creating draft orders and Capturing payments for your future reference.
Alternatively, you can also use a third party app called Split ‑ Partial Payments that allows your customers to pay for a single order in different ways. We can also reach out to the developers of this app should you have further queries about its app functionalities and features."
