Widgets: Show Recent Posts Only if the Posts Have Both Categories X and Y

you’re using 'category__and' => array($category_name) which according to the input becomes 'category__and' => array('594,3')

Here ‘594,3’ is not a valid id

instead try 'category__and' => explode(',', $category_name)

This will make the input as 'category__and' => array('594', '3')