As jdm2112 and SallyCJ said in the comments, the query lacks of 'field' => 'slug'
. So:
'tax_query' => [
'relation' => 'AND',
[
'taxonomy' => 'link-category',
'terms' => $link,
'field' => 'slug'
],
[
'taxonomy' => 'despacho',
'terms' => $despacho,
'field' => 'slug'
],
],