Sorry to read that you're having issues with registering your new PC.
TV Anywhere and Sky apps are separate platforms and have separate devices that can use these services.
If you sign into your TV Anywhere account (on a laptop/PC) on the right hand side under My Profile you will see Device Settings and Sky Movies & Sky Sports.
Device Settings will show the devices registered for TV Anywhere. Sky Movies & Sky Sports show the devices to use for Sky.
On your account there are already 2 devices registered to use Sky, if one of these are not in use you will have to remove a device then register the new PC by streaming a channel on the Sky Sports website.