I am running TC Deluxe 2016 and have external referenced some existing drawings to a new drawing.  The data from the existing drawing's model space is available at the new drawing to be set as a new view and then moved to the new drawing's paper space viewport. 

But is there a way to also move the data that was on the existing drawing's paper space also over to the new drawing's paper space?

Welcome.   

Not as far as I know,  from the help file - "Only files that have objects in Model Space can be added as Xref's."  I would think it could become confusing if multiple x-refs are added,  each with various paperspaces, layers, styles etc. Especially if like me, one uses the same viewport names, layer names etc in multiple drawings.

As a side note - one can 'add' files with objects, paper-spaces, viewports etc. into a drawing, by using the Extract From, BUT one would lose any advantage of x-refs, in that the drawing would not update if the original is altered. 

Have you tried copy & paste