Moving to a new computer should not make a difference.
Your license is linked to a specific Adobe CC account. The serial number was rejected because it was being activated on a different CC account (unless it’s a firewall or Internet filter that is blocking access to our activation servers):
Make sure you’re logged onto your old account and it should work.
If you were using a very old Creative Suite version (before Creative Cloud came along), your script would be linked to the serial number of InDesign you were using, and, again, if you’re using the same version of InDesign on the new computer, it should work.
You may need to purchase a “license transfer” in that case.
More details here Reactivation Woes, or: What to do if you’ve got a new computer and it doesn’t let you activate your product! – Id-Extras.com
Feel free to post back if you have more questions.