はい。ワシントン州の免許証は無効になります。
When you move to a new state and visit their DMV to get your new driver’s license, that new state informs your old state that you have moved and that your old driver’s license is now invalid. Your old state invalidates your license and updates your file with your new address and information from your new state.
You can only have one license from one state at a time in the US. All states inform the other states of your new status.