I'm not sure how to explain with words so hopefully the images can do it justice, but is there an easy way to essentially duplicate vertices/lines such that the new set will remain attached to faces one side while the original set remains attached to the other side?
Ah ha! Apparently it's called Rip-Fill. Simply select the desired vertices/edges and press Alt+V. And it seems you can pick which side moves by which side the mouse started on when you press Alt+V. Amazing.