stuck between two school boards

What is the name of your state (only U.S. law)? NJ

Names of districts withheld for obvious reasons.

On June 30, School A laid off all its nontenured faculty (including me).
On July 22, I accepted a job teaching at School B. I have not signed a contract yet and will not be offered a CONTRACT until the school board meeting, which will be on August 16.
On July 29, I found out (from my supervisor at school A) that my name is on the list of nontenured faculty being rehired. However, this will not become OFFICIAL until the school board meeting , which will be on August 18.
Of the two, I would prefer to work at School A (this is my tenure year). Can School B force me to take the job there?

No they cannot force you to take a job anywhere. Slavery was outlawed some time ago.

No, of course they can't force you. But they can say that a position with School B is the only position they are going to offer you. From there, the decision is entirely yours: a teaching position with School B or no position at all.