Move Label Not Working

I had the same problem and opening the layer property, I saw the coordinates of the label apearing in red :

By clicking on the menu (on the red icone) and selecting "store data in the project", I was able to solve the problem and move the labels. I hope it works for you as well !

This bug is still present in version 3.6.2.

You need to open the Layer Property, click on each of the data defined coordinates X and Y:

and select the "store data in project" box:

This is a known bug. It was reported for QGIS 3.0.3 and was fixed, but it re-appeared in 3.2.1.

Your options are:

1. Wait for a bug fix.

2. Revert to QGIS 3.2.0 - download old version of QGIS here.

3. Use a static field for data-defined label positions.

   Create fields called "LabelX" and "LabelY", with Field Calculator expressions x($geometry) and y($geometry), respectively. Set these fields to control the label position.

   With the layer in editing mode, try using the Move Label tool - hopefully it will re-position the labels and change the "LabelX" and "LabelY" values. If not, you'll have to position your labels by manually editing the values in these fields.