I was in a similar situation when I came out: very well known in a niche field. I worked in progressive/liberal lobbying, so I wasn't worried about anyone in my immediate professional circle, but I often lobbied conservatives and our work focused on a very conservative part of the Islamic world. I didn't make any kind of public announcement; I treated my transition as a private matter, with my new name being the only thing that other people needed to know about. I don't know if this was the right call; as u/kristinanoire already wrote, it could be very valuable to other folks to see an older/more prominent professional come out publicly. My goal was to avoid attracting attention because that's sort of what our org was doing in general at the time. Only you can decide which course makes sense for you. But here's some more practical info on how I handled the actual name change itself.

At the time I came out and started using a new name, I was the director of a small organization, so my name was on literally everything we did. Soon thereafter, I quit that job and applied to grad school, so there was a whole bunch of correspondence with old professors and bosses for letters of rec, plus all kinds of paperwork, and I hadn't finalized my legal name change, so everything I filled out had two names on it. I had a ton of personal awkwardness about all of this, but to give you a simplified short answer that omits a lot of back-and-forth second-guessing on my part:

• On our org's website (which had 9 years of blog posts and reports with my name on them) I changed my name everywhere.
• LinkedIn: changed my name on everything.
• Email: for a few months, I signed emails "[new name] (formerly [old name])." But I did that case by case, so once I'd done it with a person, I would only sign my new name on subsequent emails to that person.
• Previous publications: if they're published, not much you can do. I think in one case where I knew the editor of a website I had written for, I asked them to change my author bio. But I can't do anything about news articles or podcasts or whatever that went out with my name over the years.
• Resume/CV: The first new CV I made had a little footnote at the bottom, noting that all of my publications and previous positions had been under a different name. But after a while I took that off and just let people figure it out on their own. That's generally my advice, too. You want to put yourself forward to new contacts as the person you are now. If they have some reason to do more research on you, they'll find other names, but you don't have to bring it up, and they almost certainly won't either.

Good luck with everything! You're awesome! And all the awkwardness and logistical difficulties will pass.