http://talk.collegeconfidential.com/university-washington/2015670-how-hard-to-direct-transfer-into-mechanical-engineering-at-uw.html describes the OP’s situation better.
Washington public universities with ME include EWU and WSU.
There are also numerous public universities in the Western University Exchange that offer 1.5 times in-state tuition to students from other states in the region. Most of them are not that selective (the more selective ones do not need to discount out-of-state tuition to fill their seats). See http://wue.wiche.edu/search1.jsp to search for schools.