It sounds like you are doing a good job of focusing on structure over details in your initial read and that is why you are doing well on the general questions. On specific questions, you have to get back into the passage and really understand the details. When you hit a specific questions, you should know pretty quickly where to find the information in the passage. You must re-read that section of the passage to gain the detailed understanding you didn't get during the initial read. Specific questions don't require you to re-read the entire passage, just the paragraph or section of a paragraph that relates to the topic in the question.
Remember some key principles regarding RC on these specific questions. To start, your process is not to find the correct answer but rather to eliminate the wrong answers. Many people use this elimination methodology on Sentence Correction, but it's every bit as valuable on RC questions. As you are eliminating wrong answers, you should be very concerned with each word in the answer choice. Correct answers on RC can't be mostly correct. The GMAT creates good wrong answers by taking true statements and invalidating them with an incorrect term or two (ie extreme language like must, cannot, etc.). Lastly, beware of the "inference" specific questions. These inference questions require that you go back to the passage and re-read the specific section, but your answer won't be explicitly stated in the passage. Instead, the answer to these inference questions is something that has to be true based on the information on the passage.
Practice makes perfect on RC questions. Remember the techniques and get plenty of practice.
KW