As hexcoder says, you only need to read two lines at a time. Rather than using C-style loops to move along the sequence shuffling as you go you could use unpack and map; as BrowserUk has observed, there seems to be no point in shuffling more than once.

use strict;
use warnings;

use List::Util qw{ shuffle };

my $inFile = shift;
open my $inFH, q{<}, $inFile
   or die qq{open: < $inFile: $!\n};

my $outFile = $inFile . q{_shuffled.fasta};
open my $outFH, q{>}, $outFile
   or die qq{open: > $outFile: $!\n};

my $window = 10;

while ( not eof $inFH )
{
    my( $id, $seq ) =
       map { chomp; $_ }
       map { scalar <$inFH> }
       1 .. 2;
    
    my $shuffled =
       join q{},
       map { join q{}, shuffle split m{} }
       unpack qq{(a$window)*}, $seq;

    print $outFH qq{$id\n$shuffled\n};
}

close $inFH or die $!;
close $outFH or die $!;
[download]

I hope this is of interest.

Update: Corrected cut'n'paste error in script, I was erroneously unpacking $shuffled instead of $seq.