All emails to the Governor sent through her website are first viewed by employees of the Constituent Services Department within the Office of the Governor. These emails are coded with subject information and then routed within the office or to individual agencies for follow-up. All communications to the Governor are made available to her executive policy staff who use this information when helping the Governor as she moves forward with issues important to the residents of Washington State.
I hope this answers your question.
What I was looking for wasn't a direct email from firstname.lastname@example.org, but rather some sort of indication that my email wasn't going into some black hole. I'm not totally sure that if you do send anything to Dino at his email address collecting website that you'll get anything back at all.
Either way, the governor's office seems to be taking this more seriously.