If you set $/ to "" you will read paragraphs at a time. That is, you will read a block of lines that are separated by 1 or more blank lines. Then you can split on newline boundaries:

$/ = ""; # read paragraphs
while (my $para = <>)
{
  my $lastHeading;

  my @lines = split(/\n/, $para);
  if (@lines == 1) # heading?
  {
     $lastHeading = $lines[0];
     next;
  }
  
  my $hitMe;
 
  if ($lastHeading eq 'MEANINGLESS TEXT')
  {
   $hitMe = $lines[4]; # fifth line
  }
  elsif ($lastHeading eq 'ALTERNATIVE TEXT')
  {
   $hitMe = $lines[5]; # sixth line
  }
  # now do something...
}
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