Deleting old payment information

update the old to something else.  eg, if you can pick paypal, at a certain point you'll click a paypal link at which point another browser tab will open that you can cancel and save the "changed" payment info thereby removing the previous payment info.

note: if you have no payment info you will be seeing lots of alerts/reminders/nags to update your payment info.

 I can see that we can update the payment information, however, I don't see any reference to deleting the information. See the following article for details

https://helpx.adobe.com/in/manage-account/using/update-billing-information.html

If all you want is to delete the payment information then probably you would have to contact Adobe support to get an official word on how it can be done if it's possible at all. If you are a Teams/Enterprise customer contact Adobe support via the Admin Console. If you have an inidividual subscription then follow the instructions given below.
1. Go to https://helpx.adobe.com/contact.html?rghtup=autoOpen
2. Wait a bit on the page that opens up, it should automatically open a chat popup. If that does not happen then look for a call out/chat icon at the bottom right, click over it and the chat popup should open
3. In order to talk to a human type "AGENT" in the chat box
4. If you are not able to get to the chat window, then disable your adblockers, firewall, enable cookies/scripts on your browser. If it still fails then change your browser/machine/network.

P.S. Be mindful of answering anyone who sends you a private message. See the following for more information about scammers
https://tinyurl.com/y7cfrr7y
-Manan