The setcookie function prepares the HTTP cookie header to be sent at the next page load. That’s why the $_COOKIE superglobal is not updated automatically.
You could manually update $_COOKIE for use on the current request. Just put this below your setcookie() line:
$_COOKIE['mobileesp_wp_full_site'] = 'true';
By the way, I’d prefer not to work with strings like 'true', but just with integers (0 and 1). You then can simply check them like this: ! empty($_GET['view_full_site']).