Hello, this is Emily.
If you have claimed the upgrade, chance is, your license can be now on digital license.
If that's the case, on the new computer, go to Settings > System > Activation, run the troubleshooter, choose the hardware change option. When it asks you where it is installed on right now, you should choose the old device name. Then click on Activate button. And on the next screen when it asks you where to transfer it to, choose your new device name. See if that does the trick.