The Measurement widget is designed to clear the previous measurement when a tool is selected to make a new measurement.
However, we can work around this by doing the following:
1) Listen to 'measure-end' event from the Measurement widget
2) Get and store the geometry from the event object
3) Create a graphic from the event geometry
4) Add that graphic to a separate graphics layer
Showing distance on each edge of the line drawn on the map is a bit trickier.
Instead of using the Measurement widget, I think it would be easier to use the Draw widget.
Under the hood, the Measurement widget functions similarly to the Draw widget.
In short, using events, you'd want to take the geometry of a particular line, use a task to measure it, then update your label layer with the result.
draw-amd | API Reference | ArcGIS API for JavaScript
Add graphics to a map | ArcGIS API for JavaScript