Actually, you are correct.

https://www.wklaw.com/double-jeopardy-federal/

Of course, there's still the limitation that the State would need to find it's own law that the accused violated - they have no jurisdiction over Federal law (and vice-versa).

Still... it doesn't sit quite right with me. If someone is acquitted by a jury in, let's say, Rhode Island for murder. It doesn't seem right that the Federal government could step in and hold an entirely new trial thereafter. Seems to fly in the face of what Double-Jeopardy is supposed to prevent.

Isn't that exactly why federal hate-crime legislation exists? To allow a do-over in federal court if juries in some jurisdictions ignore crimes against some victims?

And for that matter, even if acquitted in a criminal trial, someone can be found guilty in a civil trial (with a lower standard of evidence). See, famously, OJ Simpson.

I suppose, but for that you cannot be sued merely for committing a crime - you need to prove tangible personal direct damages.

...and that can happen regardless of criminal conviction.