Morph Shapes in After Effects in only one direction

You have a couple of options here. The first is to use multiple shape layer paths and add the Merge Paths property and do some shape layer animations. The second is to make sure that you draw the ending shape before you set a keyframe, then move back in the timeline and edit the position of the points. Points on a vector shape (mask or shape layer) move in a straight path from one keyframe to another. You have to be. Careful when you move the points so you don't get any weird transitions. It make take setting a keyframe every few frames to get the transition you want. 

one more tip, you can place the big shape on top of the small shape, then apply linear wipe effect and animate the transition 

You likely simply need a lot more keyframes. That's just how AE masks work. Also, visually there is no need at all to animate the actual source shape. You could just have a second shape appear underneath the original one and expand from there. And of course there's a ton of utility scripts on AEScripts.com that may also help. anything more would require to see the keyframes and inbetween shapes, anyway.

Mylenium