In Shopify there's several places you can update an address:
The customer record
If you're using a subscription app, in the subscription record
Any Orders that have been created for that customer
If you update the Customer Record or Subscription Record, it won't trickle down to the Order. You'll want to update those records for future orders, but most importantly you'll want to update the address in any current Orders.
Make sure to modify the address in the Shopify Order, then click Refresh Data in the Import page in Pirate Ship to get the new address.