One thing that Visio allows is the many ways of creating the same process. For example, you indicate that if the connector line has no handle you cannot move the text. That is not true. To move ANY text, simply use the "Text Block Tool" from the standard toolbar (select the drop-down menu to the right of the "Text Tool" indicated with an "A") and right-clicking on the text...then move where desired (once you right-click you will see a dual-triangle icon that will allow the move once you hoover atop the text you want to move).

Hope this helps.