When using Aegislash's "Stance Change" Ability, do I have to switch the Doublade & Honedge underneath it too?
No, you only switch one Aegislash card for another Aegislash card.
Source: XY FAQ (2014-02-06), TPCi Rules Team (2014-02-06)

Relevant cards

Note: All applicable cards may not be listed below.
XY1_85
XY1_86