iCloud+ plan/ iCloud Storage payment

How do you change over to a new credit card as a payment method for iCloud+ plan?

Under Apple ID in the Settings App, under “Payment and Shipping” it displays my old card. When I click into it, it doesn’t show the old card but shows the new card that I’ve added. The bills are still going to the old card. I’ve signed out of my iCloud account and signed back in and it still hasn’t been resolved. I am also afraid that I will lose whatever I have stored in my iCloud during the interim.

Please assist as the old credit card belongs to a friend and the payment is going to them.