Adobe Suddenly Can't Charge My Credit Card

The "them" that you cannot reach means Adobe Customer Service?

Here is a URL for Adobe CS billing problems:

https://helpx.adobe.com/contact.html?step=CCSN_membership-account-payment_payments-invoice s-orders_stillNeedHelp

I am just another user like yourself so don't have any internal information about your situation but could it be that something has happened to the status of your credit card and maybe you can contact the bank that issued the card and see if they see anything strange.  It may be that you'll have to find another card to pay with, although it doesn't have to be pre-paid, or at least mine isn't, it's just a bank debit card from my local bank.