I am trying to create a subplot (6 plots) of world maps into which I am writing chosen shapefiles. My issue is with my placement of subplots: They are overwriting each other. I
subplot by itself will create axes that don't overlap, but will delete any existing axes that do overlap. So put the six subplot calls first, and change their positions at the end. Use set(handle1,'Position',h1pos), not subplot(...) to change the position. You can also use axes to create an axes object without deleting any existing overlapping axes. Since you're setting the position manually anyway, the subplot command has no advantage for you.
You could also consider using the new tiledlayout functionality.