...
if ($total + $member->compressedSize > $limit) {
        $zip->writeToFileNamed("zipped_$archiveCount.zip") == AZ_OK or die $!;
        $archiveCount++;
        print "Total archive size: $total bytes\n\n";
        $total = 0;
        my $zip = Archive::Zip->new;
    }
    
    $zip->addMember($member); # <---- This line
...