That seems like an interesting problem. Don't do anything in a hurry, this could become a quagmire. If you have a lot of MR points you don't want to lose them so don't go closing anything.
The first thing I'd do is sign up with Credit Karma or a similar service (making absolutely sure to get the SSN correct!) and see if your AMEX accounts have been getting reported regularly. If AMEX has your correct credit file but is telliing you your SSN shows you as deceased, you need to either verify what SSN they have (even if it is just "yes, we have the same number you do" or "no, we have a different number") or else check with Social Security to make sure your number is still valid. Personally I would check with SS first.
If AMEX has an incorrect number and an incorrect credit file (inaccessible to you) then I think you'll need to close those cards and get new ones. However, if they have the correct credit file but wrong SSN you'll need to ask them what to do--that might be particularly difficult to correct. And if they have the correct SSN but SS is actually reporting you deceased, you may have an indentity theft issue, or perhaps SS just screwed up.
I've called the SSA, they said everything is okay. AMEX checks my credit, but they do not seem to post my personal credit cards. They did before, not sure why this hasn't happened for a few years. I just really didn't take notice since I pay my balances off before the closing of the statement. It's just habit, I've do this with all banks.