I am trying to move a vertex in QGIS 3.2. I can select nodes in QGIS 3.2 (which show up blue), but every time I try to move it with the node tool it just deselects the vertex.
How do I move a node in QGIS 3.2?
All you have to do is to click once to select the vertex (node) and move your mouse to move the vertex and when you finish click again to deselect the vertex (node). The detailed explanation is in this answer: Editing vertex in QGIS 3, which I think you have seen it before.
I tested your data and everything goes well. I do not have the same issue you have. I loaded your data into the canvas, clicked on start editing, selected the node editing tool, and I could move the node without any problem.
Here is your data:
Try to load your data in a new QGIS project, and start editing again. If it did not work, then uninstall QGIS 3.2 completely from your machine and reinstall it again. The current version is QGIS 3.4.