Storyline Drag and Drop Icons are Difficult to Drag and Drop

If you are using the icons that are available within SL, you'll notice they are actually already a set of "grouped" free form objects. 

You could ungroup the icon, add in a rectangle the same colour as your background, or transparent at the back of the icon elements, regroup the elements/objects back into the icon and use it for your drag-and-drop.

1 group rather than 2 groups as you mention above.