https://support.docusign.com/guides/ndse-user-guide-conditional-fields

I can't think of a way to do this using just signature tags.  However, if you use conditional fields, you could present the user with a set of radio buttons (one radio button for each document).  Once the user selects a radio button, you could have a signature tag appear only on the document associated with that radio button.  Would something like that work?

Would this work across multiple documents uploaded into the same envelope or would they have to be combined together outside of DocuSign first?

This should work across multiple documents in the same envelope.  For example, you might first present a preliminary page (document1) with the set of radio buttons.  Radio1 could make a signature tag appear on document2, radio2 could make a signature tag appear on document3, and radio3 could make a signature tag appear on document4.  The resultant envelope will show all documents, but there will only be a signature on one.