How to switch to another payment method for an already running subcription?

A payment method is for your single account as a whole. It can't be segregated to different payment mentions for different services used by your Apple ID.

When you change your payment information, it's automatically updated for all apps and services that you use with your Apple ID. Change, add, or remove Apple ID payment methods advises: When you change your payment information, it's automatically updated for all apps and services that you use with your Apple ID (my emphasis).

The only way you achieve different payments for different bills is to add other cards then reorder them (the one on top is the one which will be used) when you have a particular service's bill coming due.

I am trying to add another payment but I'm get a response that the payment method was already added yet I do see it anywhere. It is a card that I have used before and should still be there. I can only see 1 payment method... How can I see it???

Apple has posted this support document which should help you: Change, add or remove Apple ID payment methods

1. Open the Settings app.
2. Tap your name.
3. Tap Payment & Shipping. You may be asked to sign in with your Apple ID.

To update a payment method, tap the payment method, then edit your information. You may need to enter the security code that's printed on your credit or debit card. If you see Apple Pay after the last four digits of the card number, go to the Wallet app to edit the card information.

A further section describes the same process on a Mac.

1. Open the App Store on your Mac.
2. Click your name or the sign in button at the bottom of the sidebar.
3. Click View Information.
4. Next to Payment Information, click Manage Payments. (If you're using an older version of macOS, click Edit to the right of Payment Type.) You may be asked to sign in with your Apple ID.
5. From here, you can do these things:

• • Update a payment method – click Edit next to the payment method. You may need to enter the security code that's printed on your credit or debit card.