You don't need to insert the serial number until the program is actually installed. (And on a new computer, the first step after installation will be to sign into your Adobe account.) You say you can't even download the program. So there seems to be a disconnect here. If you go to the Products page of your Adobe account, you will find a download link for your program and also the registered license number.
Are you unable to get into your Adobe account? If so, you need to contact Adobe support. You can start a chat session by clicking on the blue Contact Us button under the Help & Support menu at Adobe.com.