The standard deduction applies to your Federal tax return. Here's a TurboTax video with more information about it.
The payments that are being spread over three years from your Idaho tax return will stay within the Idaho state rules / standard deduction, etc. and it will not change due to the fact that you moved to WA.
Receiving a standard deduction every year does not wipe out your tax liability. It lowers your taxable income in the year that you file and can't be applied to a past tax arrear.
