I don't really know how to explain this. I think "as" meaning is more to "like" as what you say when something resembles something.
Answering "so" and "so that",
for example, let's say:
1. I'll do my homework first so I have to go home early.
2. I'll do my homework first so that I can have more free time later.
From the sentences above, I believe "so" means something that you need to do before you can do the first action, while "so that" is more to something that you expect to happen if you do the first action.
cmiiw