WordPress plugin ajax post parameter

In your jQuery code

 data = {
      action: 'aad_get_results',
      aad_nonce: aad_vars.aad_nonce
  };

you are passing only action and aad_nonce, you are not passing form data.

If you want pass variables, you ave to pass it with ajax, use something like:

  data = {
      action: 'aad_get_results',
      aad_nonce: aad_vars.aad_nonce,
      form_data: $('#aad-form').serialize()
  };

After that, in your aad_process_ajax function use:

function aad_process_ajax() {

  parse_str($_POST['form_data'], $form_data);
  $name = isset($form_data['name']) ? : '';
  $header = isset($form_data['header']) ? : '';
  $body = isset($form_data['body']) ? : '';
  $urls = isset($form_data['urls']) ? : '';

  // the rest of your code here

}