Update: I did end up finding a driving school that specializes in helping people with phobias. I had my first lesson about a week ago and another one coming up next Sunday. I drove normally (ish) for about an hour and then got really overwhelmed and burst into tears!
Then I was so stressed that I felt basically useless for like, three days afterwards. But we'll see how the second lesson goes! Maybe it'll get easier. If anyone else is in the Bay Area, http://www.fearlessdriver.com/ is the driving school. I really like my instructor, she's really understanding and used to be afraid of driving too, which is a nice success story to see. I'm nervous that working on this will tax my resources way too much, but at the same time it's nice to have some professional support and not just be trying to figure out how to solve this problem on my own.