design mode in dreamweaver is not an accurate representation of all modern browers. It should be used as a guide, not a production area. When you drag the "layer" (which is nothing more than a div usually with absolute positioning) it changes the position relative to the containing element your "layer" lives in. Check in code view for the element you are dragging and you'll see that there is some code, probably similar to "top:300px; left:439px;", etc. You'll need to alter these numbers and then test in your browsers to see accurate results.

Hope that helps.

For more on css positioning:
http://www.w3schools.com/css/pr_class_position.asp