This may not be helpful, as I am not familiar with Washington law, but in California, the landlord may send the deposit to the last known address of the tenant if the tenant does not provide a forwarding address. In this case, the deposit should be mailed to the unit the tenant just vacated! Doesn't strike one as an effective way to return the deposit, but that's the way it is.
It is not the landlord's responsibility to make sure that the tenant gets the tenant's mail. The law requires only that the landlord sends the mail to the last known address. If the tenant does not give the landlord any other address, the last known address for the tenant would be the address of the rented property.
If the mail from the landlord gets to the rented property and is not forwarded to the tenant, the landlord should take the letter, does not open the letter, and place it in the landlord's file for the tenant. If there is any dispute whether the landlord mailed the letter, the landlord would produce the mailed, unforwarded and unopened letter.
If you know an email address for the tenant, you should also scan the notice and sent it to the tenant's email (in addition to the postal mail).
It is only your responsibility to send the secrity seposit to the last known address which in this case appears to be your property. Keep the mail, don't open it, and if and when the issue arises, you have proof that you complied with WA law.