The standard mantra should work (as explained in perldoc CGI) when dealing with cookies.
Quoted from perldoc is an explanation to -path option:

"3. a path If you provide a cookie path attribute, the browser will check
it against your script's URL before returning the cookie.
For example, if you specify the path "/cgi-bin", then the cookie will be returned to each of
the scripts "/cgi-bin/tally.pl", "/cgi-bin/order.pl", and "/cgi-bin/customer_service/complain.pl",
but not to the script "/cgi-private/site_admin.pl". By default, path is set to "/", which causes
the cookie to be sent to any CGI script on your site."

I think that you first should try to read the values of your cookie to check if it can be read by another perl page. If it is Ok, then you should not worry about the cookie generation, but worry about your JavaScript code...

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