Adding some inferred context, but avoiding the implied slurp, the following seems to address the issue:

use strict;
use warnings;
use Data::Dump::Streamer;

my @record;
my @records;

while (defined (my $line = <DATA>) or @record) {
    my $recordStart = (! defined $line) || ($line =~ /(RECORD .*)/);

    next if ! @record and ! $recordStart;

    chomp $line if defined $line;
    if (! $recordStart || ! @record) {
        push @record, $line;
        next;
    }

    die "Corrupted record: \n" . (join "   \n", @record)
        if @record < 3;

    my $rec = trim ($1);
    my ($id, $name) = map {trim ($_)} $record[1] =~ /(\d+)(.*)/;

    push @records, {rec => $rec, id => $id, name => $name};
    $records[-1]{date} = trim ($1) if $record[2] =~ s!^.*?(\d{2}\/\d{2
+}\/\d{4})!!;
    $records[-1]{address} = trim ($record[2]);
    @record = defined $line ? ($line) : ();
}


Dump (\@records);

sub trim {
    my ($str) = @_;
    
    return if ! defined $str;
    
    $str =~ s/^\s+//;
    $str =~ s/\s+$//;
    return $str;
}


__DATA__
 RECORD 1
 ######                  Full Name 1a
                         Street Address 1a
                         City 1a                       ST1a   Zip_1a  
+              COUNTY 1a
0########                Full Name 1b
 abcABCabc    99/99/9999 Street Address 1b
                         City 1b                       ST1b   Zip_1b  
+              COUNTY 1b

 RECORD 2
 ############            Full Name 2a
 99/99/9999              Street Address 2a
                         City 2a                       ST2a   Zip_2a  
+              COUNTY 2a
0###                     Full Name 2b
 abcABCabc    99/99/9999 Street Address 2b
                         City 2b                       ST2b   Zip_2b  
+              COUNTY 2b
[download]

Prints:

$ARRAY1 = [
            {
              address => 'Street Address 1a',
              id      => 1,
              name    => 'a',
              rec     => 'RECORD 2'
            },
            {
              address => 'Street Address 2a',
              date    => '99/99/9999',
              id      => 2,
              name    => 'a',
              rec     => undef
            }
          ];
[download]

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