Try this, still using hash-of-arrays

#!/bin/perl -w
use strict;
use warnings;

# input
my %hash=();
my @filename = ('File1.txt','File2.txt');
for my $n (0..1){
  parse($n,$filename[$n]);
};

output('output.txt');

# parse
sub parse {
  my ($n,$filename) = @_;
  open my $fh,'<',$filename or die "$!";
  my $key;
  while (<$fh>){
    if (/^NAME/){
      $key = $_;
      $hash{$key}[$n] = '';
    } else {
      $hash{$key}[$n] .= $_ if $key;
    }
  }
  close $fh;
}

# merge
sub output {
  my ($filename) = @_;
  open my $fh,'>',$filename or die "$!";
  for my $key (sort keys %hash){
    if (defined $hash{$key}[0]){
      print $fh $key.$hash{$key}[0];
    } else {
      print $fh $key.$hash{$key}[1];    
    }
  }
  close $fh;
}
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