Congratulations on your integration successes!
Do you want to download envelope information or the documents from the envelopers?
Use Connect
The usual technique for integrating completed envelope transactions and documents into existing systems is to use some form of the Connect feature. Connect enables you to establish a web hook, a listener, that is called by the DocuSign platform when envelopes or individual signature requests have reached a particular status level.
When you make an Envelope request through the API, you can include a call back "Connect" url as part of the request. See the eventNotification parameter
You can also establish a Connect listener at the account level instead of including the data with each envelope creation. Your Connect listener can be configured via the API or via the Preferences section of the Classic DocuSign Web App.
Message content documentation when your web hook listener is called by Connect
Another idea is to use the Retrieve product. It downloads completed envelopers and signed documents for you, to a windows server. You can then use local processes to store the data and completed documents.
Documents from envelopes
The API method Get Document from Envelope enables you to retrieve a specific document from an envelope.
Get Envelope Documents and Certificate enables you to retrieve all of the documents.
For both of these, you'd need to first loop through your envelopes to find the ones with the completed status.