Your approach could be made to work by simply checking the last two characters:

my @set = qw( A B C D E F );
my $len = 12;
my $result = '';

while (length($result) < $len) {
     my $char = $set[ rand @set ];
     next if substr($result, -2) eq $char x 2;
     $result .= $char;
}
print $result;
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Also, there's no need to use an extra variable (like $last_char), as $result already stores all the previous characters.