An adjustable stator vane for a turbine engine includes a shaft, a flange and a stator vane body that pivots about a variable vane axis. The stator vane body extends axially between a first end and a second end. The stator vane body includes an airfoil, a cavity, and a body surface located at the first end. The cavity extends axially from an inlet in the body surface and into the airfoil. The shaft extends along the variable vane axis from the first end. The flange extends circumferentially around the inlet and the shaft, and radially from the stator vane body.