Altering search form action not working for custom search module, Drupal 6

I am altering the value of the form action for the basic search form on a Drupal site. Here is the line I added to our hook_form_alter() implementation in our custom search module:

$form_id['#action'] = 'http://mydomain.com/search/customsearchmodule/';

When I view the source for the form, the action is exactly as I have set it in the above line. The problem is, after the form has been submitted, the default search results page is displayed, not the custom search results page.

How do I set the custom search results page to display, rather than the default one?

Edit

Here is an example of my hook_form_alter() implementation:

/**
* Implementation of hook_form_alter()
*/
function mymodule_form_alter(&$form, &$form_state, $form_id) {
  if($form_id == 'search_block_form') {
    $form['#action'] = 'http://www.mydomain.com/search/mymodule/';
  }
}

Also:

I added unset($form['#submit']); to my implementation of hook_form_alter() and, after submitting the form, arrived at the appropriate URL (http://www.mydomain.com/search/mymodule), only without any data passed (no keywords).

Edit #2

I am digging around in the Drupal API and in the core search module just to see how it works. This is the core search_box_form_submit() function from our /modules/search/search.module:

    /**
     * Process a block search form submission.
     */
    function search_box_form_submit($form, &$form_state) {
      // The search form relies on control of the redirect destination for its
      // functionality, so we override any static destination set in the request,
      // for example by drupal_access_denied() or drupal_not_found()
      // (see http://drupal.org/node/292565).
      if (isset($_REQUEST['destination'])) {
        unset($_REQUEST['destination']);
      }
      if (isset($_REQUEST['edit']['destination'])) {
        unset($_REQUEST['edit']['destination']);
      }

      $form_id = $form['form_id']['#value'];
      $form_state['redirect'] = 'search/node/'. trim($form_state['values'][$form_id]);
    }

Note the line where $form_state['redirect'] is defined. If I comment this line out, then the search form will land on the search/mymodule/ page, although without any search keywords passed.

Edit – Working Code

The final solution is an altered version of nmc's answer. I had to alter his mymodule_form_submit() implementation. The function should be named mymodule_submit() and the code within the function needed to be altered as well:

function mymodule_submit($form, &$form_state) {
  // The search form relies on control of the redirect destination for its
  // functionality, so we override any static destination set in the request,
  // for example by drupal_access_denied() or drupal_not_found()
  // (see http://drupal.org/node/292565).
  if (isset($_GET['destination'])) {
    unset($_GET['destination']);
  }

  // Check to see if the form was submitted empty.
  // If it is empty, display an error message.
  // (This method is used instead of setting #required to TRUE for this field
  // because that results in a confusing error message.  It would say a plain
  // "field is required" because the search keywords field has no title.
  // The error message would also complain about a missing #title field.)
  if ($form_state['values']['search_block_form'] == '')
  {
    form_set_error('keys', t('Please enter some keywords.'));
  }
  else
  {
    $form_id = $form['form_id']['#value'];
    $form_state['redirect'] = 'search/mymodule/'. trim($form_state['values'][$form_id]);
  }
}