Because data view is tied to screen scale/resolution and layout view is tied to printed paper scale/resolution, you'll always get small shifts.
There are actually two methods of adding text boxes in layout view- In 'paper' coordinates and in 'geographic' coordinates.
Adding them as paper coordinates (as it appears you've done) is as simple as just adding a text box- This will remain fixed relative to the layout so if you move the map/dataframe, these boxes will now be out of place in terms of any underlying GIS data (i.e. this mode is used for adding items like scale bars, north arrows etc which you want fixed to the same place on the page)
Adding them as geographic coordinates will mean they move with the map (similar to annotation) and are tied to geographic locations. This is used for adding ad-hoc labels etc to your map data- To add these elements, in layout view, double click the dataframe to activate/enter it (the border will now have a diagonal hatch), then add the text box etc. Click out of the dataframe to exit this mode.
Hope that helps (if so, please mark as the answer :))
