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;
}