We moved from CA to OR in August 2018 and bought a house. How do we address moving costs? Also Do I have to claim my SSI on both state forms?

[JulieCo]

Moving costs are no longer deductible on 2018 federal or Oregon tax returns as part of the Tax Cuts and Jobs Act of 2017, unless the move was for a military change of duty station.

Your SSI is not entered on your federal return, therefore, if needed, the state tax returns will ask for it.  If required to report it, you would prorate it between the two states based on the amount of time you were a resident of each state.

Below is information on filing part-year resident state returns in TurboTax:

https://ttlc.intuit.com/replies/3302008