Once I have drawn the shape, per below, how do I simply lengthen the line?
If I try via Transform Controls, it will also increase the width of the line (i.e. It transforms the whole shape as I drag diagonally upwards).
How do I just make the line longer?
Googling helps :) But I thought I'd answer the question myself so others can find the answer.
You have to manually select the anchor points pertaining to the part of the shape you wish to resize. So, select the "direct selection" tool, click on the relevant anchor points (the small white squares on the corners of the shape, which should turn black once selected - hold shift to select more than one), then drag and resize to your heart's content.
Potential other methods: http://forums.adobe.com/thread/372453