Change your while loop:

 while (my $line = <$fh_in>)   {       
    my $text = $line;
    chomp ($text);  
    
    #Strip/remove whitespace between lines of text file;
    while (<STDIN>)
    {
      print if (!/^\s*$/);
    }
                     
    print $fh_out "$text\n";  #Save stripped results; 
         
  }
[download]

as follows:

 while (my $line = <$fh_in>)   {       
     print $fh_out $line if (!/^\s*$/);
 }
[download]

Update: Sorry, I wrote that above message on my mobile device in the train commuting back to home, the train was arriving near my town, so I copied and pasted the code a bit to hastily. The above code should be:

 while (my $line = <$fh_in>)   {       
     print $fh_out $line if $line !~ /^\s*$/;
 }
[download]

Update 2: I had not noticed when I wrote the first update above, but poj sent me a CB message proposing a very similar correction. Thanks to poj. Further update: I had also not seen that poj also posted the correction as an answer to your question, so that these updates end up not to be very useful...