Check the escape_char setting.

Adding this line after the loop shows " is the escape_char.

say $csv->escape_char;

Getting it to import also needs a few other settings, so perhaps the auto_diag process needs work, or this is just difficult data to auto diagnose.

I also re-arranged the code to work on my machine and added a use 5.010;.

use strict;
use warnings;
use 5.010;

use Text::CSV qw(csv);

my $csv = Text::CSV->new ({ 
  binary      => 1, 
  auto_diag   => 1, 
  escape_char => '\\', 
  quote_char  => "'", 
  allow_whitespace => 1
});
my $o = \*STDOUT;


# Read/parse CSV
while (my $row = $csv->getline (\*DATA)) {
    my @selected = ( splice(@{$row}, 0, 2), splice(@{$row}, -6, 2) );
    $csv->say($o, \@selected);
}

say $csv->escape_char;


exit(0);

__DATA__ 
8, 9, NULL, 'Filler', '555-999', '77:88', 0, 0, 0, 0
8, 9, NULL, 'Filler', '555-999', '77:88', 'A \' B , C', 0, 0, 0
8, 9, NULL, 'Filler', '555-999', '77:88', 0, 0, 0, 0
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