Your school probably did include utilities -- that's calculated as part of the cost of living. However, there is no guarantee that this is very realistic. Often times what they quote as rent + cost of living (I forget what term is used to bundle this all together) is the same amount as the cheapest rent I could find. Car expenses are usually set somewhere between $90-150/month -- your school probably did include them, but you may have found it not to be adequate enough.
First off, as paean alluded to, many school financial aid offices set the upper limit at which you can borrow. Usually, this is the entire budget for attending school for a year -- tuition, books, school supplies, living expenses (rent, utilities, food, etc.), some car expenses, health insurance, and some miscellaneous expenses. If you want to borrow above this maximum amount, most private lenders (like Citibank, BofA, WellsFargo, etc.) require approval from your FAO first, and most FAO's will not give this approval -- they want to make sure that you are not borrowing wildly and are keeping your debt under control. When I researched it last year, the only company I found that would lend me money without FAO approval was P.L.A.T.O -- you can find their website if you do a google search. Anyways, they had very high rates, high origination fees, etc., so I realized I might actually be better off using my credit cards than taking out such a loan.
My best advice would be to try and get a loan from your family for some of these miscellaneous expenses you mentioned, if it's possible -- hopefully they would offer you better terms than P.L.A.T.O. (that wouldn't be hard to do). If you are graduating or have a birthday coming up, and you know that people are planning on getting you gifts, try to get stuff you need, like for you apartment or room, money for a rent deposit, etc. My final suggestion would be to work your butt off before med school starts. You stated that you are living with your family now -- if you work for a couple of months, that should allow you to save most of your income since you aren't paying for living expenses. You should at least be able to come up with the money for either some of your moving expenses or rent deposit, if not both.