I know you're not allowed to get extra federal financial aid (including Stafford loans) for supporting a family. I doubt any school-based aid would work here either. The only option I can think of is private loans, which are more expensive and require a credit check.
All private lender want you to get the amount you want to borrow certified by your financial aid office as being within your budget. As they are required by federal law to not include expenses incured by your spouse, you essentially can't get additional funds for him/her. However, it is possible to for both of you to live on your student budget if you are very careful about money, get housemates, never eat out, take advantage of free entertainment opportunities, etc. Some students even manage this with kids, you just have to be very frugal.
I saw one lenders website that said they would loan something like up to $110K in private loans for med school. There is some program out there called EXCEL loans thru the private lenders. I would just do a search on the web.
What I know from undergrad and loans was that my school calculated my budget. Then they figured my contribution and told me what I could get in aid. I was then offered to take the loans out for additional aid up to a maximum limit set by the federal gov per year so even if they told me I was only eligible for $6000, I could actually take up to $10,500 (part of it being unsubsidized of course).
Private lenders may fill in the gaps in the same way but I don't know for sure. It kind of seemed like they operated using the same model from what I could find on the internet.