You can send an HTTP request without CURL using PHP5. This answer is based off of How do I send a POST request with PHP?. You'll have to enter your variables into the $data array.
$url = 'https://domainname.com/dashboard/api';
$data = array('to' => PHONE_NUMBER, 'from' => SENDER_ID, 'message' => TEXT, 'email' => EMAIL, 'api_secret' => SECRET, 'unicode' => BOOLEAN, 'id' => IDENTIFIER);
$options = array(
    'http' => array(
        'header'  => "Content-type: application/x-www-form-urlencoded\r\n",
        'method'  => 'POST',
        'content' => http_build_query($data)
    )
);
$context  = stream_context_create($options);
$result = file_get_contents($url, false, $context);
if ($result === FALSE) { /* Handle error */ }
var_dump($result);
You can test if your requests are working from your script by using http://www.postb.in or some other similar service. This allows you to see your requests to debug them (rather than relying on feedback/errors from the service you are using). Then you'll know if your requests are formatted correctly and working. Sometimes a service is down... and you spend hours troubleshooting something that's not on your end.