If the configuration for New Feature and Story is identical, then you can just edit the issue and change the issue type. But I don't think that will be the case.
You will probably have to use "move" - it will ask you for project (select the same one probably) and issue type, then ask you about any data loss/update it needs to make. You need "move" permission in the source project and "create" in the target (even if they're the same project)
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