The spinning circle could be due to software issues. A system restore (after a backup of course) should remedy the issue for you.
But before you do that:
The issue with verifying your device to use iCloud from another device is based on your security type. Whenever you use Two-Factor Authentication and you don't sign out of older devices that you no longer use, Apple assumes that you still have those devices. Or if you signed out of a device and then restored/reset it, Apple will assume the device is a new device and will ask for the old "device" aka old OS to verify the new OS to use iCloud. The only way I know to remedy this feature is to completely turn off Two-Factor Authentication until Apple makes this a much easier/tolerable issue. To turn off Two-Factor Authentication, go to www.appleid.apple.com (or www.appleid.com) and sign into your account. Once signed in, go to the Security tab -> click edit in the far right -> then turn off Two-Factor Authentication. You'll have to use Secondary Authentication as the primary security type for your account. All this means is that you'll need to use security questions and answers along with your password to protect your account.