Mr. Jackson is correct. Your handbook is generally designed for distribution among the "at-will" employees at your place of employment. Your contract will be deemed to supersede the handbook and its policies. To the extent that that two conflict, the contract is likely to win out.
The issue you may be facing is whether the school will hire you back if you refuse to sign the handbook / at-will acknowledgement (instead of the contract). If you are given a contract in addition to the new handbook and acknowledgment, you should be in a good place (because the contract will supersede the handbook). But be careful. They may try to get you to just sign the handbook / at-will acknowledgement (without a contract), meaning you would be subject to termination without cause. Be aware of your rights in this situation and if it gets to confusing, consult a local Washington lawyer; you can probably find one here on Avvo.
I hope that information is helpful.
My thoughts and comments are for general information only. The do not constitute legal advice and they should not relied upon as such. My thoughts and comments do not form an attorney-client relationship--I am not your attorney. Please seek the advice of a licensed attorney before taking any acts based upon my thoughts and comments or the comments of others.
I would not sign the handbook without first crossing out and initialing the "at-will" language. I assume you are talking about contracts and handbooks for next year. I assume also that the employer is not going to give you a contract unless you sign the handbook first. It sounds to me like you need a union. The Washington Education Association represents public school teachers but not private school teachers. They may have some helpful information for you though and they do accept associate members from private schools.