In ArcMap 10.1, I can 'Start Editing' a polygon layer and select a feature with the 'Edit tool'. If I hold down the left mouse button, I get the four way arrow cursor but then I am unable to drag the polygon anywhere. I can however edit vertices and change the relative position using the 'Move" command. Same issue with all shapefiles and projects. This functionality works fine on the different machine with ArcMap 10.1. I assume that something in the settings is preventing this type of editing.
This could be the "Sticky move tolerance". Go to Editing Options from your Editor Toolbar.
On the General tab, the second option is Sticky move tolerance.
Set this to 0.