I checked the free package you mentioned in the link you provided. It says it provides email forwarding. It doesn’t help really.
But the good part is it keeps your domain. The not-well part is it’s managed, and you may not do anything non-managed by your own, you need to accept the policy and follow their way.
I cannot validate the code you did in server php side, A php expert should help you with this.
As I just had some study over php mail function, it seems it calls a native app called sendmail, please note if your host blocks this, so you will fail.
Some stuff about your code, I’m not sure if php support this form of string concat as you did ("From: $name \n Message: $message";), maybe you were about "From: ".$name. "\n Message: ".$message; instead?! (same for "From: ".$email." \r\n";)
Also try to log any error if mail raise, the log may help you to find out the issue much better.
Please note the email you send ($email) in your code, should be subset of your domain, or be allowed send from your account(e.g. try not send from abc@gmail.com while you are not or not allowed send mail as gmail.com)
I’ve never worked with managed hosted system like your case, but I believe you can make it.
Keep going on good work! happy programming.