Yeah. I live sort of adjacent to the U District and it's definitely true there are no budget options around here. There are pre-furnished "Apodments" in a few places that are like dorms and run $600/mo or so for a bedroom, but when I looked into those I found the company to be so hellacious to deal with that I don't know if I would recommend it. Studios will run you over a grand a month and the complexes rarely have any amenities, and no utilities included typically. And utilities here are expensive.
The cost of getting a place with multiple rooms or bedrooms is so much more, splitting it doesn't even end up costing much less than renting a studio alone. The cheapest option is probably to rent a single room in a larger place from someone.
For some perspective, I have a dog so most of the cheaper places won't rent to me. A tiny bedroom sized studio in a building that accepts dogs around UW runs $1400+ plus utilities per month, and will charge $500 for the dog non refundable plus two months rent deposit to move in, plus another $25-50/mo in "pet rent." They opened up a new building here that was meant to be "affordable housing for professionals" at UW and the cheapest studios were $1450 no utilities included. So that's what's considered a bargain around there.