Do you use a fulfilment service to fulfill your orders, or do you fulfill them manually? When you open the order, does it give you the ability to mark as fulfilled? If you could post a few screenshots of what you're seeing (while blurring out any sensitive information), then I can take a look!
I manually fulfill my orders through printful. On the printful side, this order is closed out. When I open the order in shopify, the only options it gives me are: refund, duplicate, cancel, archive and view order status page. There is also an option at the bottom of the page to request fulfillment. Below is a screen shot of what the page looks like when I click on "request fulfillment".
Thank you for the screenshot! From my experience with Printful, their API should be fulfilling the orders automatically once the order has been shipped, so it shouldn't require you to mark the orders as fulfilled manually. Given that it shows request fulfillment, I suspect there's a disconnection between our the two systems.
If you click on fulfill, does it mark it as fulfilled? Once you do, does the order appear in Printful? If yes, you should be able to cancel the fulfillment from within the Printful dashboard to ensure you're not double billed. That said, I'd also recommend touching base with Printful support to confirm up why the order is stuck in the unfilled status.
If I missed anything, or if you have any questions, let me know!