I was able to successfull de-register the conflicting scripts and dequeue them, causing the errors to go away:
wp_dequeue_script('wpdreams-aspjquery');
wp_dequeue_script('wpdreams-ajaxsearchpro');
wp_deregister_script('wpdreams-aspjquery');
wp_deregister_script('wpdreams-ajaxsearchpro');
wp_deregister_script( 'jquery' );
. . .
This is not ideal as it introduces a dependency on my plugin to your plugin. I am still curious as to why the scripts aren’t compartmentalized. Looking at the high compatibility, minified code, it looks like it shouldn’t case issues as it is, in fact, namespaced.