The problem is that in your new method, $self is the name of the class and not an object. Try this:

sub new {
  my $self = bless {}, shift;
  $self->{dbh} = shift;
  # etc
  return $self;
}
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An incidental point. You should pick a consistent indentation style with an indent in the range 2-4 characters and stick with it. It significantly improves ease of comprehension. Steve McConnell's excellent book Code Complete has a lot more on this and other issues, I highly recommend it (even though it is not about Perl).