Compared to Texas, apartment rent in or near Seattle is going to be higher. Washington has no income tax, and Texas doesn't either so there's that. Washington has very high sales tax, but not on groceries.

I suggest you rent convenient to your job, because car commuting from Bellevue or Kirkland to downtown can be brutal. And if the job is not downtown, you might be better off renting close to the job.

Start learning where the Sounder trains and the light rail go. If you need to take a bus to work, a one-seat ride (no transfer needed) is the best. Apartment rent, like real-estate, varies depending on supply and demand (location) so start learning the market on-line.

A lot of variables at play here so do your research.