I second this. There is absolutely no reason you need a computer science degree to do web development work. If your work is good, there are people out there who appreciate people with broader experiences.
I may be biased, because I was an English major once upon a time. All of my training has been on the job and in my spare time. Admittedly, I had the advantage of getting started in the mid 90's, when anyone with any interest or aptitude for web development could find gainful employment.
But now I am a partner in a small web development company led by 3 people with humanities degrees. If anything, if we see a resume from someone with a psychology or philosophy degree, we are apt to be more interested, not less. I don't imagine that we are in any way typical, but there are companies like ours out there.
That is not to say that we are idealists. We want to get a sense that you can really do the work, and we're looking for real-world experience. And if you can't deliver or your code is not clean, we'll cut you loose.