Also, you don't need separate prepare/execute/finish statements for an UPDATE query. You can replace it with one single do() (the DBI method).

Alternatively, again, prepare the handle for the UPDATE query outside the loop. The code could look a bit like this:
Version 1:

    $query = q{SELECT Referrals.1, Referrals.2, Referrals.3,
                      Referrals.4, Referrals.5,
                      Referrals.6, Referrals.7, Referrals.8,
                      Referrals.9, Referrals.10, 
                      Referrals.11, Referrals.12,
                      Referrals.13, Referrals.14, States.1
                      FROM   Referrals
                      INNER JOIN States On (Referrals.10 = States.1)
                      WHERE AuditID = ?};
                         
    my $sth = $dbh->prepare($query);
    foreach my $audit (keys(%$audits))
    {
        $sth->execute($audit);
        my @assign = $sth->fetchrow_array;
        $sth->finish();
            
        $dbh->do(q{INSERT INTO report_printing_table
                   (1,2,3,4,5,6,7,8,9,10,11,12,13)
                   VALUES (?,?,?,?,?,?,?,?,?,?,?,?,?)},
          undef, $audit, @assign);
     }
[download]

    $query = q{SELECT Referrals.1, Referrals.2, Referrals.3,
                      Referrals.4, Referrals.5,
                      Referrals.6, Referrals.7, Referrals.8,
                      Referrals.9, Referrals.10, 
                      Referrals.11, Referrals.12,
                      Referrals.13, Referrals.14, States.1
                      FROM   Referrals
                      INNER JOIN States On (Referrals.10 = States.1)
                      WHERE AuditID = ?};
                         
    my $sth = $dbh->prepare($query);
    my $updh = $dbh->prepare(q{INSERT INTO report_printing_table
       (1,2,3,4,5,6,7,8,9,10,11,12,13)
       VALUES (?,?,?,?,?,?,?,?,?,?,?,?,?)});

    foreach my $audit (keys(%$audits))
    {
        $sth->execute($audit);
        my @assign = $sth->fetchrow_array;
        $sth->finish();
            
        $updh->execute($audit, @assign);
    }
[download]

Maybe, just maybe, this also fixes your problem.