It looks the endpoint is deprecated in the version 2022-04 https://community.shopify.com/topic/1564746. I guess that might be a reason of unstable service. Fulfillment Orders API to be used now, here is a guide for a new way of fulfillment https://shopify.dev/apps/fulfillment.
However in docs on 2022-04 Fulfillment API still exists
https://shopify.dev/api/admin-rest/2022-04/resources/fulfillment#post-fulfillments, however now it seems to require providing fulfillment_order_id.