If you want some substantial space for $2k or under, I would look at the East Bay exclusively.
Be prepared to be shocked either way.
https://angel.co/jobs?utm_source=home_gs#find/f!%7B%22locati...
If you really have the chops, and know how to stick to your guns and position yourself in negotiations, companies are willing to break the $150K barrier (even the $200K barrier). They're simply hoping you're not smart enough to know how.
This includes not advertising the fact that they're willing to go above that barrier.
Fortunately they aren't the only choices. I work at a pretty big and well known company (not Google, Facebook, or the like). My salary is about what you list, and I'd hazard a guess that I could get something comparable by hinting at the army of recruiters around here I'd be interested in changing jobs, while avoiding startups.
The reality is that for the value even average software developers bring to the companies they work for, the average salary for the average developer in the SF Bay area ought to be at least $150,000/year. Top notch engineers and computer scientists working on cutting- or bleeding-edge systems and research ought to be north of $200,000 on average.
It basically sounds like "Come work at your dream job, where you competed in a global talent pool to get, for employers that conspire to suppress wages, where you can't actually buy any real estate, and we will pay you just enough to survive"