Check if Post Title exists, Insert post if doesn’t, Add Incremental # to Meta if does

This would need a query.

So building on your code:

<?php
$postTitle = $_POST['post_title'];
$submit = $_POST['submit'];

if(isset($submit)){

    global $user_ID, $wpdb;

    $query = $wpdb->prepare(
        'SELECT ID FROM ' . $wpdb->posts . '
        WHERE post_title = %s
        AND post_type = \'stuff\'',
        $postTitle
    );
    $wpdb->query( $query );

    if ( $wpdb->num_rows ) {
        $post_id = $wpdb->get_var( $query );
        $meta = get_post_meta( $post_id, 'times', TRUE );
        $meta++;
        update_post_meta( $post_id, 'times', $meta );
    } else {
        $new_post = array(
            'post_title' => $postTitle,
            'post_content' => '',
            'post_status' => 'publish',
            'post_date' => date('Y-m-d H:i:s'),
            'post_author' => '',
            'post_type' => 'stuff',
            'post_category' => array(0)
        );

        $post_id = wp_insert_post($new_post);
        add_post_meta($post_id, 'times', '1');
    }
}

Should do it