You are very close. To query each nameserver once, put your nameservers in a list and iterate through the list, assigning each nameserver once to a resolver and querying it. See the following:

use Net::DNS;
use strict;

my @nameservers=("192.168.1.1", "192.168.2.2", "192.168.3.3");
my $nameserver;
foreach $nameserver(@nameservers){
  print "checking on $nameserver\n";
  my $res = new Net::DNS::Resolver;
  $res->nameservers($nameserver);
  my $query = $res->search("www.perlmonks.com");
  if ($query) {
    my $rr;
    foreach $rr ($query->answer) {
      next unless $rr->type eq "A";
      print $rr->address, "\n";
    }
  }else {
    print "query failed: ", $res->errorstring, "\n";
  }
}
[download]

If you assign multiple nameservers to one resolver, it will use them as alternate nameservers in case it doesn't hear back from one of the others.