The output record separator may help you do what you want:

$\ = $endofline;
print OUTFILE $processed_string; # Appends the wanted end of line corr
+ectly
[download]

This may be a problem if you are not writing to OUTFILE only while $\ is changed. If you can limit the change to a defined scope (ie: localize) that would probably be better:

{
  local $\ = $endofline;
  my $old_handle = select OUTFILE;
  # Write to OUTFILE only
  while (<>)
  {
    process($_);
    print $_; # write to selected handle (OUTFILE) and append $\
  }
  select $old_handle;
}
# $\ has its old value again
[download]