Random note: you might want to look at the File::Spec module instead of dealing with path separators directly. Also, the File::Basename module might come in handy.

WRT your problem, couldn't you just add a couple of lines to your process_dir() routine that checks to see if the file is there and if so, slightly modifies the output filename? Something like this (untested):

use File::Basename;
use File::Spec::Functions;
# ...

while (my $file = readdir DIR) {
   # ...
   # $newfile contains the full path to the new file
   my $extra = "a";
   while (-e $newfile) {
      my ($path,$basename,$suffix) = fileparse($newfile,'.rtf');
      $basename .= $extra;
      $extra++;
      $newfile = catfile($path,$basename,$suffix);
   }
   copy($oldfile,$newfile) or die;
}
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