function tj_process_existing_paragraphs($content) {
    $content = wpautop($content);

    $content = preg_replace('/<p>/', '<p class="tj-paragraph-style">', $content);

    $content = preg_replace('/<p class="tj-paragraph-style">\s*(?:&nbsp;)*\s*<\/p>/i', '', $content);
    $content = preg_replace('/<p>\s*(?:&nbsp;)*\s*<\/p>/i', '', $content);

    return $content;
}

remove_filter('the_content', 'wpautop');
add_filter('the_content', 'tj_process_existing_paragraphs', 10);