The MCAT is incredibly time intensive, and you are not at all alone in not finishing the sections. If you are getting a good percentage of the question you answer right, then you just need to work on speed. The key to this is to work as many Full Length Exams as you can. They start considering the details and implications of the topics, and end up off track, without even realizing it.
Intellectual curiosity is a wonderful thing.
However, on the MCAT, you need to be able to turn it off. Just get through the passages as quickly as possible, grabbing the important facts as you go. Remember, the MCAT is an open book test. If you need to clarify something in a question, the passage is right there to go back to. Personally, what worked for me was to force myself to just read word after word, even if it didn't make any sense, to just keep going no matter what.