Sounds like a perfect use for the flip-flop operator.

use strict;
use warnings;

while (<DATA>) {
  next unless /^internal name/ .. 0;

  next unless /^need this/;

  # your code here...
  print $_;
}

__DATA__
a line
need this (not really)
another line
internal name
another line
a further line
need this
another line
need this
[download]

Note that by using a zero as the righthand operand, we get a flip-flop that turns on but then never turns off.