Try using the cookie jar like this:

use LWP::UserAgent;
use HTTP::Cookies;

my $WWWAgent = LWP::UserAgent->new();
my $cookies=new HTTP::Cookies(file=>'./cookies.dat',autosave=>1);
$WWWAgent->cookie_jar($cookies);
...
[download]

with the HTTP::Cookies module. This hooks the cookies onto the LWP::UserAgent. So whatever you do with that LWP::UserAgent from then on will carry those cookies with it.