Here, where public interest in the problem has been expressed, I think it's good for science to make known the fact that you believe you have solved the problems and provide some concrete details of what you have achieved. If you have reason not to show your hand right away, I think the goal should be to figure out what you are now comfortable saying in public, and how to present it. So I argue against private email.
Something I've seen some people do is write research bulletins, a bit like personal technical notes, that summarise findings on a topic in some degree of rigour, but without attempting to be comprehensive in the way that makes writing papers for peer-reviewed publication so time-consuming. The progress reports that Harvey Friedman used to send to the Foundations of Mathematics mailing list would be a good exemplar of that kind of thing, e.g., Self-contained posting 82: Simplified Boolean Relation Theory.
Starting such a bulletin series to deal with this, and then posting an excerpt here seems like a good strategy, since it allows you to quickly identify your achivements, while being in control of what details you make public.
I don't recommend regular blog posts for this, since they carry some unwanted associations, that they are conversational, open to revision, and not properly referenceable documents. Writing in a form for publication on Arxiv would make sense, but a Research Notes section on your publications page linking to an html page would work.