public Geometry Shape {get; set;}

await QueuedTask.Run(() =>
{
  //Don't use 'Shape'....Shape is the bounding box of the annotation text. This is NOT what you want...
  //
  //var insp = new Inspector();
  //insp.Load(annoLayer, oid);
  //var shape = insp["SHAPE"] as Polygon;
  //...wrong shape...

  //Instead, we must use the AnnotationProperties

  //annoLayer is ~your~ Annotation layer
  //at 2.x - var insp = new Inspector(true);
  var insp = new Inspector();
  insp.Load(annoLayer, oid);

  AnnotationProperties annoProperties = insp.GetAnnotationProperties();
  var shape = annoProperties.Shape;
  if (shape.GeometryType != GeometryType.GeometryBag)
  {
    var newGeometry = GeometryEngine.Instance.Move(shape, 10, 10);
    annoProperties.Shape = newGeometry;
    insp.SetAnnotationProperties(annoProperties);

    EditOperation op = new EditOperation();
    op.Name = "Change annotation angle";
    op.Modify(insp);
    op.Execute();
  }
});