You don't need to have a Wireless Network Adapter in both computers, you can mix "wired" and "wireless" connections to a router, which is probably more common than fully wireless networking.
<<<but when I tried to transfer the settings with a Flash USB, I got a message saying that because there was no wireless adapter in the PC, it couldn't transfer the settings.>>>
Not quite sure what you are doing here. When setting up Internet Connection Sharing (which you *DON'T* do when using a router) you are prompted to transfer settings via floppy, flash disk or whatever. I have never actually discovered the purpose of this and certainly never done it. However, as you have Internet access already working, you don't want to change your networking setup.
You do need to ensure that both computers are in the same Workgroup and any firewall software needs to be configured to allow access to the networked computers in its "trusted"area.