The following code fragment is something that works for me ..

use LWP::UserAgent;
use HTTP::Cookies;
use HTTP::Request::Common;

{
    my $ua = LWP::UserAgent->new;
    $ua->cookie_jar( HTTP::Cookies->new );

    my $webPage = "https://noii.nasdaqtrader.com/";

    my $res;

    #  Log in.

    {
        my $uri = URI->new($webSite);
        $uri->query_form(
            'txtUserName' => $username,
            'txtPassword' => $password
        );

        $res = $ua->get($uri);

        die "Unable to log in: " . $res->status_line
          unless ( $res->status_line =~ /200 OK/ );
    }

    #  Continue with logged in page .. 
}
[download]

Also, note that a 302 is not really an error, it's more of an informational message. It means that the web server is re-directing you to another page.