Presumably the name is always the same within the block for that account. Try

#!/usr/bin/perl
use strict;

my $acct;
my %data = ();
my %name = ();

my $infile = 'Facs_Data.txt';
open my $fh,'<',$infile 
     or die "Could not open '$infile' : $!";

while (<$fh>){
  next unless /\S/; # skip blank lines
  if (/Acct:(\d+)/){
    $acct = $1;
  }
  if (/Name:([^\s]+)/){
     $name{$acct} = $1 unless defined $name{$acct};
  }
  push @{$data{$acct}},$_ if ($acct);
}
close $fh;

for my $acct (keys %data){
  my $outfile = join '_',$name{$acct},$acct,'2017.txt';
  print "Creating $outfile\n";
  open my $fh,'>',$outfile 
    or die "Could not open '$outfile' : $!";
  for (@{$data{$acct}}){
    print $fh $_;
  }
  close $fh;
}
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