I seriously doubt you will get a helpful response to your question. What you have posted is far to "mocked up" and "simplified" for anyone to be able to guess what you are actually doing.

Simple example. You say:

Instead, I'm getting:

command: cd hello
command: cd world
response:
command: cd goodbye
...
[download]

But looking at the snippets you've posted, the first thing I notice is that the print statement that would be producing that response: trace is   print "resp: $responseData\n";, making your "actual output" nothing of the kind.

In addition, the logic of your sub:

sub sendData {
  my $request = $_[0];
  $responseData = ""; #class member var
  print SOCKET "$request\n" . "\000" or die "print SOCKET: $!";
  
  $responseData = <SOCKET>;
  ...
  print "resp: $responseData\n";
}
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is such that there should be no way for your sends and receives to get out of sync as you follow each transmission with a blocking read.

Unless of course you have set the socket to be non-blocking, in which case, you should not be using the line-oriented diamond operator.

Add to that this line:

  $responseData = ""; #class member var
[download]

which suggests that you think you are writing OO code but obviously are not, and that you are relying upon globals vars to convey data between subroutines, and it suggests that you have one unholy mess on your hands.