I fixed a few minor errors in your script. Try this and see if it helps:

#!/usr/bin/perl

package Bloggie;

use base qw(CGI::Application);

use strict;
use warnings;
use CGI::Carp qw/fatalsToBrowser/;
use CGI::Application::Plugin::TT;
use DB_File;

sub setup {
    my $self = shift @_;
    $self->start_mode('start');
    $self->mode_param('rm');
    $self->run_modes(
        'start'            => 'start',
        'new post form'    => 'new_post_form',
        'new post success' => 'new_post_success',
        'redirect'         => 'redirect',
    );
}

sub cgiapp_get_query {
    require CGI::Simple;
    CGI::Simple->new();
}

sub cgiapp_init {
    my $self     = shift;
    my $dataname = "data.dbm";
    my %database;
    die("Can't open ${dataname}: $!\n")
      unless dbmopen %database, $dataname, 438;
    $$self{'DATA'} = \%database;
}

sub start {
    my $self         = shift;
    my $templatename = 'start.tt';
    my %database     = %{ $$self->{'DATA'}; };
    my $posts        = $database{'POSTS'};
    my $vars         = { 'posts', $posts };
    return $self->tt_process( $templatename, $vars );
}

sub new_post_form {
    my $self         = shift;
    my $templatename = 'new_post_form.tt';
    return $self->tt_process($templatename);
}

sub new_post_success {
    my $self         = shift;
    my $templatename = "new_post_success.tt";
    my (%database) = %{ $$self->{'DATA'}; };
    my @posts = ();
    my $post;
    if ( defined $database{'POSTS'} ) {
        @posts = @{ $database{'POSTS'}; };
    }
    my $query   = $self->query;
    my $title   = $self->param('title');
    my $content = $self->param('content');
    $post = {
        title   => $title,
        content => $content
    };
    my $vars = {
        title   => $title,
        content => $content
    };
    push @posts, $post;
    $database{'POSTS'} = \@posts;
    return $self->tt_process( $templatename, $vars );
}
1;
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