Move label in QGIS?

From your screenshot I suspect that you are setting the data defined x and y to the expressions "x" and "y", rather then directly binding them to the field itself.

Make sure you use the "attribute field" submenu in the data defined popup menu and select your x and y fields from that menu, rather then using expressions.

An appreciation for the answer: if the labels have rules, the button to move the label remain greyed out. Better labels without rules.