Why not just the following?

        #!/usr/bin/perl
        use strict;
        use warnings;

        use CGI;
        use CGI::Session;

        my $cgi     = new CGI;
        my $session = new CGI::Session();

        print $session->header();

        $session->param('name', $cgi->param('name')) if ($cgi->param("name"));
        $session->clear('name')                     if ($cgi->param("c"));

        if ($cgi->param("m") eq 'show') {
          print "You said:", $session->param("name"), " ", $cgi->a({href=>'?c=1'},'Start Over');
        } elsif ($session->param("name")) {
          print $cgi->a({href=>'?m=show'},'See what you said');

        } else {
          print $cgi->startform, $cgi->textfield('name'), $cgi->submit('Submit'), $cgi->endform;
        }