custom field not saved

You are wrong, your custom field is saved but you won’t see the value pre-selected in the select box. To see it you have to set the selected attribute of the select element. You can do it easily with selected() function. You should also escape the attribute values and, in your case, use wp_reset_postdata instead of wp_reset_query:

if($my_custom_query->have_posts()) : ?>

    <select name="news_one" id="news_one">

        <?php while($my_custom_query->have_posts()) :

             $my_custom_query->the_post();

             $title = get_the_title();

             ?>

            <option value="<?php esc_attr_e($title); ?>" <?php selected(  $text_field_one, $title ); ?>><?php echo $title; ?></option>

        <?php endwhile; ?>

<?php endif; ?>

<?php wp_reset_postdata(); ?>