Download the installer after logging in here:

https://store.stardock.com/login.aspx?ReturnUrl=/myaccount/products

Either download it on the new computer or save it to an external drive you'll be able to access on the new one.

Best thing to do is deactivate the license on the old computer.  Open Deskscapes configuration, click on the round question mark in the upper right of the main window, click on "change license" in the dropdown menu.  Click on "Deactivate Local Machine" in the new window.

You could also then uninstall DeskScapes from the old machine, but since you're going to wipe it there's really no point.