I don't think you can drag a copy to a new layer as you are creating it (it's definitley counter intuitive). When you make the selection of the object you want to copy you are activating the layer it's on.
I think I would drag the copy and leave it selected, then move it by dragging the little square on the right of the object or layer name in the layers panel (depending on if the layer is expaneded) onto the correct layer.