Talk to the Sheriff and ask why they won't comply with state law. If they won't follow state law, then file for a Writ of Mandamus per RCW 9.41.0975 which is a court order requiring the Sheriff to comply with the state law. Or take the easy way out and get a money order from Wal Mart.
Actually, the King County Sheriff will probably tell you that they do accept cash - but only at the courthouse:
Still seems like they are violating state law, though.Fees may be paid by personal check or money order at all locations; cash is accepted at the Courthouse location ONLY.