If anyone else reads this, I have a working theory that I haven't tested. Salesforce has a flag in Setup>Salesforce Files of "Upload Notes and Attachments as Salesforce Files" or something like that. I belive that should work for Docusign's completed documents. That being said, I have not rested it yet.

Scott, I just tried your suggestion, with no luck.  Manually uploading an Attachment does get reflected as a File, but via a DocuSign transaction it didn't work. I'm assuming it's something to do with the difference between a user action and an API call. 

@Greg That's a shame. Seems like a pretty big oversight by Docusign in my opinion. Thanks for trying it out, Greg!

I found this support link that seems to be partially correct: first, after toggling back to DS Classic I went to Connect, found Salesforce, and selected "Use Content Files" in the system settings. Next, in "Configure Login" I couldn't actually apply the custom (My Domain) URL as instructed, so I just changed  SOAP Web Services API Version  to 39. Seemed to do the trick. 

In the end, even though the exact instructions didn't work for me I  did get completed docs to show up in Files. Also: while we're not using Lightning my sandbox did have a My Domain registered. 

Greg, you're the best! Thanks for sharing your solution with me. I also put in a case with Docusign so if they provide anything simpler, I will let you know!

Haha so did I. If their response is useful I'll repost. 

Hi! We're struggling with the same issue,  but we were permanently transitioned to the new DS experience and can no longer access the DS Classic interface. Also, the support link listed does not seem to take me anywhere. Any help?

Dakota, I'm not sure what happened with that hyperlink in my original post. Here's a shortened link: http://bit.ly/2KvmMMs 

If you still need to try to resolve this you can open a case with Docusign to have them temporarily open access to Classic. 

Hi Dakota,

I was able to get this fixed by logging a case with DocuSign and they were able to temporarily move us back to DS Classic, where I made the change, and then they switched us back.  Hope that helps!

This is still an issue today, I followed the steps in this doc: https://support.docusign.com/en/articles/DocuSign-Connect-for-Salesforce-Writeback-to-Files-in-Lightning and it still did not work. A call to DocuSign support is still needed. They had to toggle something from their end and then it worked.