Ideally a model is made in such a way that the steps are well organized, and changing the size is just a mater of editing a sketch dimension. This model is pretty hopeless. The irony is that it probably would have taken less time to model with solid modeling practices, and resulted in a model that takes no time at all to make size changes.
moving on
at this point direct modeling is going to be your best bet, so using the move is fine. as you have found there is something about the geometry that interferes with moving faces. here is a screen cast of a heavy handed method of splitting the body, moving it, and connecting it back up.
https://knowledge.autodesk.com/community/screencast/372d15e6-8820-4a64-acce-46855a8737e8
I noticed the fillet around the edge when all the way across the vertical face and consumed it, so I tried editin that fillet in the timeline so it had a smaller radius. After that, moving faces the way you were trying to do worked as expected. here is a screen cast of that-
https://knowledge.autodesk.com/community/screencast/696d38db-f0fa-4886-9217-032dcdf02c09
@jeff_strater -is the way the filleted edge interferes with moving a bug or limitation? definitely weird.