public string TextString {get; set;}

await QueuedTask.Run(() =>
{
  //annoLayer is ~your~ Annotation layer...

  // use the inspector methodology
  //at 2.x - var insp = new Inspector(true);
  var insp = new Inspector();
  insp.Load(annoLayer, oid);

  // get the annotation properties
  AnnotationProperties annoProperties = insp.GetAnnotationProperties();
  // set the attribute
  annoProperties.TextString = "Hello World";
  // assign the annotation proeprties back to the inspector
  insp.SetAnnotationProperties(annoProperties);

  //create and execute the edit operation
  EditOperation op = new EditOperation();
  op.Name = "Update annotation";
  op.Modify(insp);
  if (!op.IsEmpty)
  {
    var result = op.Execute(); //Execute and ExecuteAsync will return true if the operation was successful and false if not
  }
});