Query where ANDing slug values not working

You can do this using the WP APIs like this:

$args = [
    'post_type' => 'post',
    'tax_query' => [
        'relation' => 'AND',
        [
            'taxonomy' => 'your_taxonomy_name',
            'field'    => 'slug',
            'terms'    => array( 'arizona', 'speech-language-pathology' ),
        ]
    ],
];
$query = new WP_Query( $args );

Then you can use $query as you would any standard post loop.

If you only want the IDs you can use the fields option of WP_Query, but keep in mind you won’t gain all the caching and performance benefits you might have gotten otherwise, especially if you’re going to pass the ID into get_post

$args = [
    'post_type' => 'post',
    'tax_query' => [
        'relation' => 'AND',
        [
            'taxonomy' => 'your_taxonomy_name',
            'field'    => 'slug',
            'terms'    => array( 'arizona', 'speech-language-pathology' ),
        ]
    ],
    'fields' => 'ids'
];
$query = new WP_Query( $args );
$posts = $query->get_posts(); // $posts is an array of post IDs