I'm getting support requests from users about searching, they don't realise the "clear search" button is there so get confused when their search is added to their previous one instead of being a new search.
Is there a setting in the admin panel for it that I'm missing? I tried looking

insert this into the beginning of your search.php script
$_zp_current_search->clearSearchWords(); to clear the search.

When I add that it makes no search results show up

You can try the development nightlies, then. There is added a parameter to the printSearchForm() function to control this behavior.