With HTML::TokeParser::Simple

(assumes all links within class="full")

#!/usr/bin/perl

use strict;
use warnings;
use HTML::TokeParser::Simple;

my $html = do {local $/; <DATA>};
my $p = HTML::TokeParser::Simple->new(\$html)
  or die "can't parse: $!";
  
my ($in_full, @href);

while (my $t = $p->get_token){
  
  next if  
    $t->is_start_tag('div') 
      and 
    $t->get_attr('class') 
      and 
    $t->get_attr('class') eq 'content';
    
  $in_full++, next if 
    $t->is_start_tag('div')
      and
    $t->get_attr('class') eq 'full';
     
  $in_full = 0, next if
    $t->is_start_tag('div')
      and
    $t->get_attr('class') ne 'full';
        
  next unless $in_full;
  
  push @href, $t->get_attr('href') if 
    $t->is_start_tag('a');
  
}

print "$_\n" for @href;

__DATA__
<!-- some other divs and so here -->
<div class="full">
    <div class="content">
        <ul class="topics">
            <-- I want extract these links div class "full" only -->
            <li><a href="foobar">foobar</a></li>
            <li><a href="foobr2">fobar2</a></li>
            <li><a href="fobar3">foobr3</a></li>
        </ul>
    </div>
</div>
<div class="otherclass">
    <div class="content">
        <ul class="topics">
            <-- I DO NOT WANT these links -->
            <li><a href="fbaor">fbaor</a></li>
            <li><a href="fabar2">fabar2</a></li>
            <li><a href="fbar3">fbar3</a></li>
        </ul>
    </div>
</div>
<!-- some other divs and so here -->
<div class="full">
    <div class="content">
        <ul class="topics">
            <-- I want extract these links div class "full" only -->
            <li><a href="foobar4">foobar4</a></li>
            <li><a href="foobr5">fobar5</a></li>
            <li><a href="fobar6">foobr6</a></li>
        </ul>
    </div>
</div>
<div class="otherclass">
    <div class="content">
        <ul class="topics">
            <-- I DO NOT WANT these links -->
            <li><a href="fbaor">fbaor</a></li>
            <li><a href="fabar2">fabar2</a></li>
            <li><a href="fbar3">fbar3</a></li>
        </ul>
    </div>
</div>
<!-- some other divs and so here -->
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