my $cookies = HTTP::Cookies->new();
my $ua = LWP::UserAgent->new( cookie_jar => {} );
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ICBW but it looks to me like you are setting up a new cookie jar in $cookies but then failing to associate that with your user agent. The docs for LWP::UserAgent also suggest using a HTTP::CookieJar::LWP instead. Try replacing these lines with

my $cookies = HTTP::CookieJar::LWP->new;
my $ua = LWP::UserAgent->new( cookie_jar => $cookies );
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and see if that gets you any further.