append code after the_content not working

previous_post_link and next_post_link both output the link directly, which won’t work in your case because you’re trying to assign the result to a variable. Use get_previous_post_link() and get_next_post_link() instead-

function add_pagin( $content ) {
    if ( is_singular('post') ) {
        $content .= get_previous_post_link() . get_next_post_link();
    }
    return $content;
}
add_filter( 'the_content', 'add_pagin' );