Here is a function that keeps HTML tags in tact, adds a "Read More" link at the end of the excerpt and trims the excerpt after the first paragraph.

if ( ! function_exists( 'wpse0001_custom_wp_trim_excerpt' ) ) : 

    function wpse0001_custom_wp_trim_excerpt($wpse0001_excerpt) {
    global $post;
    $raw_excerpt = $wpse0001_excerpt;
        if ( '' == $wpse0001_excerpt ) {

            $wpse0001_excerpt = get_the_content('');
            $wpse0001_excerpt = strip_shortcodes( $wpse0001_excerpt );
            $wpse0001_excerpt = apply_filters('the_content', $wpse0001_excerpt);
            $wpse0001_excerpt = substr( $wpse0001_excerpt, 0, strpos( $wpse0001_excerpt, '</p>' ) + 4 );
            $wpse0001_excerpt = str_replace(']]>', ']]&gt;', $wpse0001_excerpt);
            
            $excerpt_end = ' <a href="'. esc_url( get_permalink() ) . '">' . '&nbsp;&raquo;&nbsp;' . sprintf(__( 'Read more about: %s &nbsp;&raquo;', 'pietergoosen' ), get_the_title()) . '</a>'; 
            $excerpt_more = apply_filters('excerpt_more', ' ' . $excerpt_end); 
            
            //$pos = strrpos($wpse0001_excerpt, '</');
            //if ($pos !== false)
            // Inside last HTML tag
            //$wpse0001_excerpt = substr_replace($wpse0001_excerpt, $excerpt_end, $pos, 0);
            //else
            // After the content
            $wpse0001_excerpt .= $excerpt_end;
        
            return $wpse0001_excerpt;   

        }
        return apply_filters('wpse0001_custom_wp_trim_excerpt', $wpse0001_excerpt, $raw_excerpt);
    }

endif; 

remove_filter('get_the_excerpt', 'wp_trim_excerpt');
add_filter('get_the_excerpt', 'wpse0001_custom_wp_trim_excerpt'); 

Please note, the "Read More" will be in a new paragraph, if you need the "read More" next to the last word, just remove the // from this code

//$pos = strrpos($wpse0001_excerpt, '</');
            //if ($pos !== false)
            // Inside last HTML tag
            //$wpse0001_excerpt = substr_replace($wpse0001_excerpt, $excerpt_end, $pos, 0);
            //else
            // After the content