Grabbing $_POST data Background

A quick summary of where I stand right now:

We have a low traffic site here that was redone in Drupal about 8 months ago, before I was in the picture. An external design firm sold the owners on the technology as a way to enable non-programmers to update site content easily. Unfortunately, they did it wrong - they built our theme in-place over the Bartik theme instead of subclassing it, things like login boxes are written directly in the Basic Page Body textareas that our non-programmers are supposed to be updating, on and on and on beyond what's relevant to this post. Had I been here when it was forked over, I'm not even sure I would have taken delivery.

I've been meaning to pick up Drupal one of these days to go in and try to refactor this hot mess, but the non-Drupal work that actually keeps us all employed around here has prevented that.

Fast forward to now. I've got a feature request for the Drupal site that they want ASAP. It's a very simple application. Given the rush, I decided to develop the application as a straight PHP app to monkey patch onto the Drupal site. Based on some forms I had to maintain within the Drupal frontend (not using webforms, mind you) that I am able to $_POST back to some non-Drupal PHP with, I thought I could get away with this.

Now as I am trying to integrate my super vanilla PHP code into our Drupal environment I am running into all sorts of problems. At first my $_SESSION array wasn't persisting from page to page but I was able to figure that part out on my own by googling.

TL;DR: The question

Where I am stumped is with my $_POST array. I POST form data (an honest to God, completely naive HTML form) from one non-Drupal .php file to another non-Drupal .php file and it never arrives. The right way to do this - and the way it will get redone someday - is clearly to use the provided Drupal APIs for building forms.

Unfortunately, I am in such a position that there is absolutely, positively no time to rewrite this the right way today. With that in mind, is there a way to pass this POST data directly to the php file I am using for form processing?

EDIT:

It looks like maybe I am getting redirected internally and the data is not coming along for the ride. Could this be it? If so, it looks like Forms specified using the API Drupal provides can do something like :

$form['# redirect'] = false;


to prevent their being redirected. Is there some perhaps hidden input field I could pass in with my non-Drupal forms to accomplish the same thing?
To tackle your exact question, let me first tell you that what happens in the form is the correct thing. 

When you submit the form, it takes data, validates it and calls appropriate submit handlers to take care if the rest. 
The form, once its routing is completed, does a 302 redirect to the same location so a browser F5 hit does not resubmit data. 
It's a general practice to avoid double submissions. 

You will need to make a module that alters the form using hook_form_alter and alter the form. 

You can change form's action URL to the other PHP file or add an extra submit handler that posts data to the other raw php file using drupal_http_request. 

You will find sanitized and structures data in $form_state['values'] (second param of a submit handler is usually called $form_state). 
$_POST is also available at this stage.