Hey,
So you need to first work on successfully setting the browser cookie. Then you may find the rest of your code works for you.

So I think the problem is that you are misusing the -path parameter when creating a new cookie.

From the perldoc for CGI::Cookie: (emphasis my own)

-path points to a partial URL on the current server. The cookie will be returned to all URLs beginning with the specified path. If not specified, it defaults to '/', which returns the cookie to all pages at your site.

It is NOT where the cookie gets stored. It is specifying which URLs get the cookie. I'd delete it, and see if that helps.