Thomas, did you sign in on the second PC with the same LiveID that you used on the first PC?
On the first PC, open OneCare and click Help and then About to see your Subscription ID. That is the LiveID to sign in with. If so, can you sign in with that LiveID at http://billing.microsoft.com to see the status of your subscription?
I don't understand why support was unable to help you (they can't activate your PC manually), but then I don't understand why you would be asked to pony up for a new subscription if you signed in with the *same* LiveID on PC #2 that you signed in on PC #1 with.
-steve