Honestly, I think you should ask your school counselor why the numbers look like this on your transcript. It doesn’t make sense that a student with a 3.44 UW and six completed AP classes should have an weighted GPA only .02 points higher.
It sounds as if you feel like the unweighted average is accurate though, correct? The question is just why the weighting doesn’t raise it more.
UW engineering is going to be pretty reachy since you’re out of state. Even if you get into the university, there’s a good chance you won’t get direct entry to the engineering school. Getting into an engineering major without starting out in the engineering school is nowhere near a sure thing.
UW out of state, probably without the maximum OOS merit grant, would be quite expensive, too. What’s your budget?
What is it that you like about UW in particular? Do you specifically want to be in the Pacific Northwest? Do you just want to be far from home, with different terrain and climate? Where else are you planning to apply?