Each of your code blocks will overwrite the previous, you need to add to the array rather than replace it:
$args['tax_query'][] = array(
'taxonomy' => 'region',
'field' => 'id',
'terms' => $_POST['region_filter']
);
relation
is AND
by default, so you can just omit it, but if you wanted to explicitly set it:
$args['tax_query']['relation'] = 'AND';