I installed DBD::mysql 4.018 and DBI 1.616 and ran the following based on your posting (note I had to invent $planname and substitute $dbname. Also I took out the fetch as it cannot work:

use strict;
use DBI;

my $h = DBI->connect('dbi:mysql:database=test', "xxx", "yyy");

eval {
    $h->do(q/drop table CUSTOMER_PLAN/);
};
$h->do(<<EOT);
CREATE TABLE `CUSTOMER_PLAN` (
   `RowId` int(11) unsigned NOT NULL auto_increment,
   `Name` varchar(6) default NULL,
   `Credits` smallint(6) unsigned default NULL,
   `Price` decimal(8,2) default NULL,
   `Credit_Value` decimal(3,2) default NULL,
   `Status` char(1) default NULL,  # A=Active, I=Inactive
   `Display_Order` char(3) default NULL,
   `Description` varchar(255) default NULL,
   PRIMARY KEY (RowId),
   KEY planname (Name),
   KEY descript (Description(15)),
   KEY dispord (Display_Order)
 ) ENGINE=MyISAM DEFAULT CHARSET=utf8;
EOT

$h->do(q/INSERT INTO `CUSTOMER_PLAN` VALUES(1, 'Free',     0,  0.00, 0
+.75, 'A', '001', 'Try us out - receive 3 free recipes.')/);
$h->do(q/INSERT INTO `CUSTOMER_PLAN` VALUES(2, 'Shine',   12,  9.75, 0
+.75, 'A', '002', 'Shine with a small introduction to Vibrant Healthy 
+Living.')/);
$h->do(q|INSERT INTO `CUSTOMER_PLAN` VALUES(3, 'Seeker',  35, 18.75, 0
+.75, 'A', '003', '<i>-noun.</i> An adventurous person who is seeking 
+new experiences and new information to create a higher level of under
+standing and enlightenment.')|);
$h->do(q|INSERT INTO `CUSTOMER_PLAN` VALUES(4, 'Sage',    60, 47.00, 0
+.75, 'A', '004', '<i>-noun.</i> A generous, lively person on the jour
+ney for the possession of wisdom and experience. This person is energ
+etic, vibrant and famed for health and wisdom.')|);
$h->do(q|INSERT INTO `CUSTOMER_PLAN` VALUES(5, 'Guru',   120, 89.00, 0
+.75, 'A', '005', '<i>-noun.</i> A divine teacher. This person has vas
+t knowledge and practices vibrant living while sharing with their com
+munity to create a greater understanding and manifestation of truth a
+nd beauty.')|);


sub db_err
{
  my ($pmsg, $psql, $perrnum, $perrstr) = @_;
  print "<p>There was a problem: $pmsg.</p>\n";
  print "<p><i>Attempted command:</i> $psql</p>\n";
  print "<p><i>Error:</i> $perrnum - $perrstr</p>\n";
}

my $dbname = 'test';
my $planname = 'Free';
my $sql = "SELECT DCredits, Price, Description FROM $dbname.CUSTOMER_P
+LAN WHERE Name='$planname'";
my $sqlh = $h->prepare($sql) || db_err("preparing statement handle", $
+sql, $DBI::err, $DBI::errstr);
$sqlh->execute || db_err("executing query", $sql, $DBI::err, $DBI::err
+str);
#####while (@row = $sqlh->fetchrow_array())
#####{
#####  ($credits, $price, $descript) = @row;
#####  # process the data on each loop here
#####}
#####$sqlh->finish;
$h->disconnect;
[download]

It fails with:

DBD::mysql::st execute failed: Unknown column 'DCredits' in 'field lis
+t' at /home/martin/930402.pl line 45.
<p>There was a problem: executing query.</p>
<p><i>Attempted command:</i> SELECT DCredits, Price, Description FROM 
+test.CUSTOMER_PLAN WHERE Name='Free'</p>
<p><i>Error:</i> 1054 - Unknown column 'DCredits' in 'field list'</p>
[download]

When I rerun it with DBI_TRACE=15 I get the following near the end:

    -> prepare for DBD::mysql::db (DBI::db=HASH(0x9f482a0)~0x9f483c0 '
+SELECT DCredits, Price, Description FROM test.CUSTOMER_PLAN WHERE Nam
+e='Free'') thr#9dd4008
    New 'DBI::st' (for DBD::mysql::st, parent=DBI::db=HASH(0x9f483c0),
+ id=undef)
    dbih_setup_handle(DBI::st=HASH(0x9f48180)=>DBI::st=HASH(0x9e50408)
+, DBD::mysql::st, 9ece010, Null!)
    dbih_make_com(DBI::db=HASH(0x9f483c0), 9f3e808, DBD::mysql::st, 25
+2, 0) thr#9dd4008
    dbih_setup_attrib(DBI::st=HASH(0x9e50408), Err, DBI::db=HASH(0x9f4
+83c0)) SCALAR(0x9ecdf30) (already defined)
    dbih_setup_attrib(DBI::st=HASH(0x9e50408), State, DBI::db=HASH(0x9
+f483c0)) SCALAR(0x9ecdfb0) (already defined)
    dbih_setup_attrib(DBI::st=HASH(0x9e50408), Errstr, DBI::db=HASH(0x
+9f483c0)) SCALAR(0x9ecdf70) (already defined)
    dbih_setup_attrib(DBI::st=HASH(0x9e50408), TraceLevel, DBI::db=HAS
+H(0x9f483c0)) 0 (already defined)
    dbih_setup_attrib(DBI::st=HASH(0x9e50408), FetchHashKeyName, DBI::
+db=HASH(0x9f483c0)) 'NAME' (already defined)
    dbih_setup_attrib(DBI::st=HASH(0x9e50408), HandleSetErr, DBI::db=H
+ASH(0x9f483c0)) undef (not defined)
    dbih_setup_attrib(DBI::st=HASH(0x9e50408), HandleError, DBI::db=HA
+SH(0x9f483c0)) undef (not defined)
    dbih_setup_attrib(DBI::st=HASH(0x9e50408), ReadOnly, DBI::db=HASH(
+0x9f483c0)) undef (not defined)
    dbih_setup_attrib(DBI::st=HASH(0x9e50408), Profile, DBI::db=HASH(0
+x9f483c0)) undef (not defined)
    -> dbd_st_prepare MYSQL_VERSION_ID 50083, SQL statement: SELECT DC
+redits, Price, Description FROM test.CUSTOMER_PLAN WHERE Name='Free'
    >- dbd_st_free_result_sets
    <- dbd_st_free_result_sets RC -1
    <- dbd_st_free_result_sets
>count_params statement SELECT DCredits, Price, Description FROM test.
+CUSTOMER_PLAN WHERE Name='Free'
    <- dbd_st_prepare
    <- prepare= DBI::st=HASH(0x9f48180) at /home/martin/930402.pl line
+ 44
    >> execute     DISPATCH (DBI::st=HASH(0x9f48180) rc1/1 @1 g0 ima10
+41 pid#19440) at /home/martin/930402.pl line 45
    -> execute for DBD::mysql::st (DBI::st=HASH(0x9f48180)~0x9e50408) 
+thr#9dd4008
 -> dbd_st_execute for 09ece030
    >- dbd_st_free_result_sets
    <- dbd_st_free_result_sets RC -1
    <- dbd_st_free_result_sets
mysql_st_internal_execute MYSQL_VERSION_ID 50083
>parse_params statement SELECT DCredits, Price, Description FROM test.
+CUSTOMER_PLAN WHERE Name='Free'
        --> do_error
Unknown column 'DCredits' in 'field list' error 1054 recorded: Unknown
+ column 'DCredits' in 'field list'
        <-- do_error
IGNORING ERROR errno 0
 <- dbd_st_execute returning imp_sth->row_num 18446744073709551614
    !! ERROR: 1054 'Unknown column 'DCredits' in 'field list'' (err#0)
    <- execute= undef at /home/martin/930402.pl line 45
DBD::mysql::st execute failed: Unknown column 'DCredits' in 'field lis
+t' at /home/martin/930402.pl line 45.
    -> $DBI::err (*) FETCH from lasth=HASH
    <- $DBI::err= 1054
    -> $DBI::errstr (&) FETCH from lasth=HASH
    >> DBD::mysql::st::errstr
    <- $DBI::errstr= 'Unknown column 'DCredits' in 'field list''
[download]

What does your trace output look like on the failing machine?