Best to ask the questions in the Washington forum, but I will answer them here.
The law says, if you do not have a Washington ID, or if you haven't resided here for 90 days, they can take 60 days to issue. Most agencies will not take that kind of time because the process is exact for both resident and non-residents, though a few do.
The preemption situation in Oregon is not the same as Washington's. In Washington, without a concealed pistol license, you can still open carry statewide. Seattle, Olympia, and other cities are not allowed to pass "loaded carry" bans like they can in Oregon.
Here's a little trick that I know that may be able to get you considered "resident" besides the utility bills. Do you have a passport? Put on your current residence address in Olympia on the app, and present a US passport for ID purposes. Most will not question further, but again, most agencies don't actually make you wait longer just because you have an out of state ID anyway.
If you have problems applying, please PM me. If you need assistance in applying for an Oregon Concealed Handgun License, please also PM me. I can assist you with the application process.