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Tailsfan101 wrote: My HS coach seems to think it's allowed, and she's very knowledgeable about SO. While soinc.org says teams may only compete in one Division at a school, my coach says that's per competition, and you can change the roster between competitions.

You can change the roster between competitions. That's fine, but is different from having the same student compete on two different teams in a season.
The national policy is here: https://www.soinc.org/start-team/team-size-grade-levels

It contains the statement "Any student can compete at only one Division level (either Division B or C, not both) at a single school."

The Idaho state director has said it's allowed for me to switch Divisions between the two competitions.

Anyway, here are the full scores:
Division B
Division C

Tailsfan101 wrote:The Idaho state director has said it's allowed for me to switch Divisions between the two competitions.

Anyway, here are the full scores:
Division B
Division C

Still, I'd ask the national committee just in case. Your state director isn't part of it (according to https://www.soinc.org/about/national-committees) so they might not be the last word. Don't get me wrong, I want you to go to nats, but I don't want a whole team being disqualified.
Ask chalker or bear, they're easily accessible.