I have a similar issue. Point and Text graphics are not eligible to be scaled/rotated, and there is no "highlighting" of any graphic if you only turn on MOVE from the Edit Toolbar.
As a result, though their edit tools have been activated, there is no flag of this to the user. Of course, this could be worked around, but it seems like a feature that has any edit tool turned on ought to give some visual clue innately, much as polygons/polylines/multipoints already do for tools such as rotate and scale.
Haven't implemented it yet in my own code, but here is one idea for working around this. Add MouseOver events for the graphics that are eligible to be edited. Upon mosueover, use the getCurrentState() method of the editorToolbar to determine if the graphic in question has a tool (i.e. MOVE) activated. Then at that point, you could change the cursor to something that denotes this graphic can be dragged around the map at leisure. (For example, that cursor with the 4 arrows pointing out from center, kinda like a compass).
Hope this helps...it is my hope that this functionality will be fleshed out within the API down the line.