Yes, you should have saved the avast license file, license.avastlic, copy that to a USB stick and use that for the new system.
Uninstall the copy on the old system, that way you only have the 1 installation.
If you didn't save the license file - Resend avast License
http://www.avast.com/resend-license-paid.php, you will need to enter the email that you used when purchasing the license, e.g. the one that the original email with the link was sent.
A Merry Christmas to you too.