You’d have to federate the change out to other servers, editing the original author to be the new account. You’d need all the different activitypub platforms to support this otherwise different sites would show different info.

It depends on the goal. If you just want to have a record of all your old posts, then it’s just a matter of making sure there’s a copy on your new server and associating them with your account. You don’t even have to federate the change necessarily. If you want to change the ownership of all the original posts, that’s a lot harder and may not be possible. There’s a separate a issue of what happens when a server disappears, but that has a whole lot of related issues, like communities disappearing, which are a problem. But if we assume it’s just about a user wanting to move from one server to another when the old server is not shutting down, it becomes a bit simpler.

And it needs to originate from the original server as you couldn’t trust a new server claiming it’s you.

Yes, which shouldn’t be a problem if the old server isn’t shutting down.