My issue has now been resolved by iTunes support.
They will not tell me what they did, but I think it was a total account reset. They then told me to not enter any credit card information, but to download a free item from the store from each of my devices so they became 'trusted' and only then to enter my credit card info.
Note, however, that I was not getting the credit card declined error, I was getting an unable to verify your account error.
I think success is highly dependent upon what support 'specialist' you get. I was being given the run around until someone else picked up the case and escalated it.
I think, unfortunately, the best approach is to keep trying, including submitting multiple support requests, until the 'right' person responds.