You should keep the form on new-customer.php and set the form action to POST to another script (something like new-customer-process.php) and then paste your PHP code into that script. Essentially the user won’t know any different as they will be redirected back to the home page.
To give you a little more information, the form is what is sending the headers which is why you need to separate out the form and the processing script. The redirect needs to be BEFORE any HTML is sent.