The simplest answer is that you are the copyright holder, and Stack Exchange's license to your work is not exclusive, so you have unlimited rights to use your own work however you please. You cannot go wrong republishing your own work, since you are still the copyright holder of the work and never promised an exclusive license to anyone (e.g., as you might if you were a professional writer submitting a piece to a magazine or newspaper).
So, you technically have no obligations to yourself or to Stack Exchange. The interesting question here is how things would go if you staunchly refuse to identify yourself as the author of the Stack Exchange post, even privately to Stack Exchange. What could Stack Exchange do, in response to this apparently-unauthorized (but really in secret, self-authorized) reuse of Stack Exchange content?
Insofar as Stack Exchange doesn't hold copyright on the post, they can't take any action against you for copyright violation. Furthermore, even if they wanted to treat it as some kind of terms-of-service violation, Stack Exchange can't prove that your license to use the post didn't come directly from the original author (e.g., maybe this reposter is a friend of the author and has never even heard of Stack Exchange), or that the reposter is in fact the original copyright holder (as is the case here).
In an academic context, you may sometimes have an obligation not to self-plagiarize, but this is an academic ethical concern, not a legal issue. (Nor does it appear to be relevant in this particular, non-academic scenario.)