Just for context, almost every federal court is a branch of a state court.

This is not true at all.

Federal courts are part of the judicial branch, not the executive branch. So they don’t shut down when the executive branch “shuts down”, because the shutdown laws don’t apply to them. As a practical matter federal courts can keep running for a while using saved up court fee revenue. They will eventually run out of that money and gave some tough choices about what to do.