On initial inspection, I strongly suspect that the problem you describe with larger input file is because of your profligate use of memory pushing your process into swapping and thus slowing it down by a factor of 1,000 times or more.

An example. In this loop:

foreach my $len(@uniques) {
    push @distrbtn, $len;
    my $index = 0;
    my $count = 0;
    while( $index <= $#sorted ) {
        if( $len == $sorted[$index] ) {
            $count++;
        }
        $index++;
    }
    push @distrbtn, $count;
}
[download]

You build an array by alternately pushing the length; and then the count of sequences of that length.

And then as soon as the loop finishes you convert that array into a hash:

my %dstrbtn_hash = @distrbtn;
[download]

And (AFAI saw), the array is never referenced again, but sticks around using memory for the rest of the script.

Why build the array only to convert it into a hash? Why not just do:

my %dstrbtn_hash;
foreach my $len(@uniques) {

    my $index = 0;
    my $count = 0;
    while( $index <= $#sorted ) {
        if( $len == $sorted[$index] ) {
            $count++;
        }
        $index++;
    }
    $dstrbtn_hash{ $len } =  $count;
}
[download]