has_term() need third parameter to specify which post type have this term. So, your code would be
if ( has_term('sold', 'category', 'property')) {
?>
<div class="sold_prop_note">
<h3>This Property is sold. Take a look at our current exclusives! <a href="https://wordpress.stackexchange.com/properties">View Exclusives</a></h3>
</div>
<?php
}
Beside that approach, you can use following
global $post;
if ( ( $post->post_type == 'property' )
&& has_term( 'sold', 'category' )
) { ?>
<div class="sold_prop_note">
<h3>This Property is sold. Take a look at our current exclusives! <a href="https://wordpress.stackexchange.com/properties">View Exclusives</a></h3>
</div>
<?php }