How do I accept Bitcoin Cash without using a third-party?
Your Bitcoin cash server may be a target for many since it has all the private keys.

I would suggest you get one private master key and public key.

use your public key to generate addresses and your private key to spend Bitcoins.

You can createrawtransaction and all the other commands using PHP or javascript without the need to have a running server. 

these libraries are helpful: bitcoinlib-js for javascript and bitcoin-php for PHP

You may need a server just for broadcast a transaction if you want to send bitcoins. you could use your own server or use public servers like electrum servers or blockchain.info