my $FILENAME4 = "organized.txt";
open(DATA2, $FILENAME4);
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You should always verify that the file opened correctly.

my $FILENAME4 = "organized.txt";
open DATA2, '<', $FILENAME4 or die "Cannot open '$FILENAME4' $!";
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#remove all previous re-organized files
my $remove_reorganized = "re-organized.txt";
if (unlink($remove_reorganized) == 1) {
                        print "Existing \"re-organized.txt\" file was 
+removed\n";
                          }
     
#now make a file for the ouput
       my $outputfile = "re-organized.txt";
           if (! open(POS, ">>$outputfile") ) {
            print "Cannot open file \"$outputfile\" to write to!!\n\n"
+;
                exit;
        }
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If you open the file for output instead of append then you don't have to delete the file first as that is a side effect when you open for output.

# make a file for the ouput
my $outputfile = "re-organized.txt";
open POS, '>', $outputfile or die "Cannot open file '$outputfile' to w
+rite to because: $!";
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$organized =~ s/(\w+)[^(^\d+)(\s)]/z/g;
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You are using a regular expression that says: match one or more word characters followed by a single character that is not the character '(' or '^' or any digit or '+' or ')' or any whitespace, which does not make sense.    It could be that you do not understand how character classes work?