Too right! That oughta learn me.

Rereading, I now see the approach of the OP. Perhaps (again untested):

$/ = "\$\n";
open BIGFILE, yada yada or die;
while (<BIGFILE>)
{
  my ($filename, $guts) = split /\n/, $_, 2;
  open SMALLFILE, ">$filename" or die;
  print SMALLFILE $guts;
  close SMALLFILE;
}
close BIGFILE;
[download]

or even

use File::Slurp qw( write_file );

$/ = "\$\n";
open BIGFILE, yada yada;
while (<BIGFILE>) {
  write_file(split /\n/, $_, 2);
}
[download]