Can I, the landlord be responsible for an overdue utility bill that's under the prior tenant's name?

RCW 35.21.217 gives the city a lien on the property for light and water. If your tenant does not pay, the city can proceed to collect against you as the property owner. IN 2010, this law was amended to allow landlords to receive notice from the power and water companies when the tenants become delinquent in their payments, but only if the landlord validly requests such notice (see RCW 35.21.217(3)). If your lease provides that the tenant is responsible for the utilities, then you can sue the tenant for any unpaid amounts that you are required to pay on their behalf.

The account goes to the person, so unless it is in your name, you should not be responsible.