Tech Support does not handle activation. There is a separate number for that, which should be accessible through the activation dialog, but you can try 1-866-772-3623 whih seems to be the number for CS3 and earlier.
You wrote that you “transferred files from Old Computer to New Computer.” I assume that you didn't attempt to reinstall the Adobe Creative Suite on the new computer but somehow expected the software just to work as-is.
Unfortunately, it doesn't work that way due to various software activation mechanisms that go beyond the existence of files copied from your old system.
I recommend that you totally fully uninstall CS4 from your new computer and fully reinstall it using your original media. That will allow the software activation mechanism to be properly installed on your new computer and for the software to properly activate.
Yes I finally did do that and it did work.
Thanks for your reply.