Relies on well-formed HTML:

#!/usr/bin/perl

use strict;
use warnings;

use HTML::TokeParser;

my $doc = do { local $/; <DATA> };

my $p = HTML::TokeParser->new( \$doc ); 

while ( my $outer = $p->get_tag("div") ) {
    
    next unless $outer->[1]{class} eq "full";
    my $nested_div = 0;

    while ( my $inner = $p->get_tag ) {
    
        # keep count of nested divs
        $nested_div++ if $inner->[0] eq "div";
        $nested_div-- if $inner->[0] eq "/div";

        # "full" div has closed
        last if $nested_div == -1;

        print $p->get_text, "\n" if $inner->[0] eq "a";
    }
}

__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 -->
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