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You can change the card/billing details of your Amazon account from online (using your computer). Follow the steps below (from Amazon)
On your computer, go to Manage Your Content and Devices page -----> https://www.amazon.com/mycd
From Settings, select Edit Payment Method below Kindle Payment Settings.
Select the desired credit card information, or add a new card, and then click Continue.
Enter the desired billing address, and then click Continue. You will be taken back to the Kindle Payment Settings page where you can view your edited 1-Click payment method.
When you change the card details from there, the card details on the Kindle will also be changed automatically.
If the changes doesn't reflect on your Kindle, you will have to de-register the Kindle and re-register it. Go to Settings> device > Registration and de-register from there. Then re-register it from the same menu.
If you have any questions or would like any additional information, please don't hesitate to ask. If you have any difficulty on following the steps, please let me know.