The first thing to notice when you select a shape with AutoLayout off is that, when you click the shape, it doesn't have the customary gray-dot selection. Its selection mode is the same as for AutoShapes (as in the picture on the left)—a finely dotted border with sizing (white circles) and adjustment (yellow diamond) handles.
Move a shape
Select it and drag it, as shown in the picture. The connector travels with it. (If you were to do this with AutoLayout on, the shape would simply snap back into its original position.)
Add a shape
Select the related shape first, then click Insert Shape on the Org Chart toolbar. Notably, the inserted shape doesn't snap into a designated layout position (an effect of no AutoLayout). It appears just under the related shape. Drag it into position.