while(@row = $sth->fetchrow_array) {
   ++$duplicate;
}

##</code><code>##

my $dbh = DBI->connect($dsn, $usr, $pwd, {RaiseError => 1});
my $sth = $dbh->prepare("insert into Customers(Names, ...) values(?, ...)");
eval {
   $sth->execute($new_name, ...);
};
if($@) {    # Got an error!
    # The insert failed - probably due to the duplicate key constraint
    # You can check the $@ variable which holds the error
    # string that caused the insert to fail to see what
    # error happened.
    # Handle this the same way that you would handle the
    # case where your checkDuplicateName() returns true
 }