It's generally illegal for public employees or officers of any kind to make more than incidental use of public resources for any personal, rather than public-business purposes (political or otherwise), and further its even more illegal for public officials to use public resources for political campaign purposes.

So, yeah, using a non-city email address as his campaign contact email seems not only legal and ethical, but probably legally and ethically mandatory.

EDIT: the applicable Washington state statute on political campaign uses of public resources is: https://app.leg.wa.gov/RCW/default.aspx?cite=42.52.180

The main body of the rule is "No state officer or state employee may use or authorize the use of facilities of an agency, directly or indirectly, for the purpose of assisting a campaign for election of a person to an office or for the promotion of or opposition to a ballot proposition. Knowing acquiescence by a person with authority to direct, control, or influence the actions of the state officer or state employee using public resources in violation of this section constitutes a violation of this section. Facilities of an agency include, but are not limited to, use of stationery, postage, machines, and equipment, use of state employees of the agency during working hours, vehicles, office space, publications of the agency, and clientele lists of persons served by the agency."

EDIT AGAIN: Looking at that again, that is specific to state officers or employees, so may not be applicable to city officials, but represents the kind of rule that is commonly in place.