This is a classic problem, and the classical solution is to have another method in the base class which returns the $db value.

  package IDObject;
  my $db;
  sub set_db {
    $db = shift;
  }
  sub get_db {
    return $db
  }
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Then in the derived class you simply do this:

  sub fix_it {
    my $this = shift;
    my $what = shift;

    my $results = $this->get_db->search($what);
    .
    .
  }
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