If you are not running a bitcoin core node, then I don't think there is a need for a new computer just for this. It would be better though to move your BTC from the wallets whose private keys are you going to use in the new Bitcoin Cash Wallets, as these are pretty new and there can be many unknown vulnerabilities to it.
There is no such thing as paranoia in crypto. When I install Bitcoin Cash to my dump my coins, im not going to install that software on my main machine at all. If you have a laptop, then use the laptop, since most of the time you aren't on the laptop, or you shouldn't at least use it as the main computer.
If you dont have any spare computers, get a virtual machine and install an OS and install the software there.
Call me paranoid but that is my advice.