How do I unlink my payment method from spouse's storage plan charges. we have separate Apple IDs and no family sharing.

Cancel your card so it no longer works. You cannot affect what is done with another person's account. In other words, calling Apple and asking them to remove it is not something they are likely to do since they have no idea you aren't somebody just trying to play a trick on somebody or that you really even are the owner of the card account.

This is a user to user forum so we can't speak for Apple.

Since you say she will not change it on her account then you really have no option but to do as I suggest and stop the card from working. You said she won't change it so the by-the-book official method that was recommended, of her changing it, clearly is not a workable solution in this case. (Frankly I would have suggested the standard method myself but I saw you had said she won't change it which is why I stepped in with a solution that will actually resolve the problem instead of the standard Apple solution.) Basically what she is doing now is using your card without your permission which is essentially card fraud. You don't need to make it a big legal issue but your card company should be prepared to re-issue you with a different card which will quietly end the issue.

I presume you have removed her name from the card account?

Sure, you could contact Apple about this but I very strongly doubt they will do anything to an Apple account that isn't yours. If they won't then you need to use a creative solution such as what I suggested.

She needs to change her payment method at appleid.apple.com