OK. Then you may have two different accounts - one is your Amazon Prime account, the other is your Kindle's registered account.
Would you like to try to get your credit card updated on your Kindle's account (assuming it not to be the Amazon Prime account)? if so, here are instructions:
The change of credit card information is actually done in your online Amazon.com account (which your Kindle uses, of course... but the c.c. info is not stored on your Kindle, so it is a matter of your internet account, really.
To change your credit card info in Amazon:
1. go to https://www.amazon.com/gp/css/homepage.html
2. and select Your Account if needed, and login if needed.
3. Then, in Your Account (or 'My Account' it might read, sometimes) select Payment Options.
4. Then select the old card that you want to change, and instead of deleting it, just choose EDIT.
5. Then enter the new card information there, carefully, and save the changes (by clicking CONFIRM, usually, or perhaps SAVE) after entering all the new data for the new card.