To make the search results appear on your facet page, add a search facet to the page and then just add these to your functions.php:

• YOUR_FACET_SEARCH_WIDGET_NAME – with the name of your facet
• YOUR_RESULT_PAGE – with the slug of you facet page

/**
 * use facetwp results as results for normal search form
 */

add_filter(
    'get_product_search_form',
    function ($form) {
        $form = str_replace('name="s"', 'name="YOUR_FACET_SEARCH_WIDGET_NAME"', $form);
        $form = preg_replace('/action=".*"/', 'action="/YOUR_RESULT_PAGE/"', $form);
        return $form;
    }
);

The same thing can be done for the normal WordPress search form

add_filter( 'get_search_form', function( $form ) {
	$form = str_replace( 'name="s"', 'name="YOUR_FACET_SEARCH_WIDGET_NAME"', $form );
	$form = preg_replace( '/action=".*"/', 'action="/YOUR_RESULT_PAGE/"', $form );
	return $form;
} );