Just hold your bitcoins in your own wallet. Why the hell would you ever do anything other than that? You hope an exchange gives you the other coins you would be owed hut in the end they don't have to give them to you if they choose not to. Just withdraw now ffs. This can't be the whole bitcoin cash debacle again.
from my whole understanding of bitcoin cash i would have needed another computer to claim them if i held bitcoin myself? i only own one computer.
there are always a lot of different "options" you can choose from to do this! you don't necessarily have to have two computers.
Option 1: use a virtual machine.
i use VirtualBox, it can be installed on any Operating System. and using it, you can create a virtual computer completely with everything. you install another OS in that virtual machine and use that as your second computer.
Option 2: use a live Linux
this is my favorite. you can use something like Tails so that it is designed not to keep anything after your session is over. but others do the same too if you know how. in any case you just download the OS (for free) and burn it on a DVD, boot up, install the wallet temporarily or rather run it and send your transactions and after you are done, you turn off you machine and everything will be gone.