A helpful clue would be to look at the URL in the address bar of your 404 page. A reliable approach for form action in WP looks like the snippet below.
<form method="post" id="thisform" action="<?php bloginfo('url'); >/dbsearch/">
I would create your page in WP, ensure the page slug is ‘dbsearch’ as this determines your permalink (http://domain/dbsearch) and the template used. You can of course use any slug you like but match this to your form action. Rename your script ‘page_dbsearch.php’ to ‘page-dbsearch.php’. WP will pick this up as the page template for your results page.
All of your form data will be available in the $_POST array to write your DB query with.