A.T.
I work part time in a financial aid office and schools do handle disbursements differently. Once you get your Student Aid Report from FAFSA (Department of Education) it will give you an amount that you are eligible for (not awarded). That information is then sent to the schools you have added to your list, the school codes you put on your FAFSA application.
Once the schools get that information they must verify that information unless you are automatically packaged and awarded. If you have been selected to be verified (this is chosen by FAFSA at random, typically 30% of students). You will have to provide copies of your tax return or other documents. Once the counselor has done this process, you will receive an award letter. You must accept this award at the school you are attending since you cannot receive pell grants or stafford loans from more than one institution.
Schools disburse students money differently but on the award letter it will state how this is handled. As far as the school that I work, the tuition is paid first, then about 5 days later book money is sent to the student on the schools credit/debit card and this amount varies since awards vary. The school does not give you the remaining amount until you have been "certified" or marked as attending classes, usually by the 12th day of class. The awards are based on a full time schedule so it sounds like you are half time at 6 hours so you would receive half of your award.
If you are not able to get a hold of someone at your school, you may want to speak with them in person. I know we have more than 100 messages just on the main line to return each day, plus the students that are waiting in line before we even get there each day. I had the same issues with the department and did not understand -- now that I work there I am amazed at how busy we are.
I know this is long but hope to have answered your questions.