If your gift option keeps turning off, goes to Draft from time to time, or switches off again after you publish it, something in your Shopify store is changing the connected gift product status.
When the gift product is set to Draft or Archived, or removed from the Online Store sales channel, Wrapped sets the Gift Option to Draft so shoppers are not offered a gift that cannot be purchased (or give it away for free).
Wrapped never changes the gift product status on its own.
The usual cause is a Shopify Flow workflow
Many stores run a Flow workflow that hides or drafts products, for example when they go out of stock, importing catalogs from drop-shipping services, or external inventory management systems.
Gift products often show zero or negative inventory because their inventory is not tracked, so these workflows catch them.
Each time you republish the gift option, the workflow turns it off again.
Fix: exclude the gift product from the workflow
In Shopify admin, open Apps, then Flow.
Open Recent activity to see which workflow ran, then open that workflow. Look for workflows that set products to Draft or remove them from sales channels.
Add a condition so the workflow skips the gift product. Do not turn the whole workflow off, it is doing its job for your other products.
In Wrapped, open Gift options and press Publish.
If you do not use Flow, another app or a bulk edit can do the same thing. The gift product's history in Shopify shows which app made each change.
Active and Unlisted are both fine.
The gift option only turns off when the gift product is Draft or Archived, or leaves the Online Store channel. Unlisted is the recommended status.