Hi Cory!
I recently ran into this issue and you are right, Duplicator has nothing to with it…I don’t like finger pointing but it is W3 TOTAL CACHE’s fault.
It adds the following line to the wp-config
define(‘COOKIE_DOMAIN’, ‘youroldsite.com’); // Added by W3 Total Cache
Apparently the search replace is unable to modify this section. So to fix it just open the wp-config and replace it with the name of your new site.
Hope it helps.