In fact the back button is not so easy to fix. Because when you perform a search currently it's a "POST" and not a "GET".
So there is solution like ask to cache page (but that not the best solution. Another solution could be to perform POST and redirect to a GET like this you will be able to perform the back to result page.

This is something we are thinking of, but no easy fix for this

This is not real a bug. it's just how browser works when trying to coming back to a POST page.
So avoid it, could you try :
edit the default.php of the module and replace <form ... method="post"> by "<form ... method="get">

Hello,

Sorry for the late replay with the simple search module.
In the setting of the simple search module, Choose 'No' to 'Display the search textfield' and in 'field 1' write the name of your title field (ad_headline if you didn't change the default title field).

The issue you have is because the textfield shown by the simple search isn't focused on the ads title but will search for the string inside the title and description by looking for each words individualy.
The advanced search form will look for the whole string only in the title field.

Best regards.