Regardless of your EFC, you can get loans, federal up to 38,500 and beyond that private loans. The maximun loan amount is set by your school, so if they say the student budget is 28K, you take all that out in Federal loans, and if they say 51K, you take out the max federal plus 11,500 private.
If your EFC is 0, you are in the same boat with almost all single medical students in their 2nd year and beyond. Whether you get grants or loans with good terms really depends on your school. Unlike undergrad, many schools will expect you to take the whole, or almost the whole amount out in loans. So it's not an entrance to a magical club with big grants as the door prize. At UCSF (entering with an EFC of 0) I received about 2500 in grants, and have taken about 30K in loans. The only difference between me and someone with an EFC of 10K is I got an 800 federal loan reserved for the poorest students, and I got a small grant.
Regarding savings, you absolutely can keep a small cushion, but when you fill out the FAFSA, it will count against you every year. So you may have to take out loans to cover the amount you have in savings, and depending on whether your student budget is over or under 38,500, they may be at poor interest rates. Although if you read my response to the 0 EFC question, you may have already realized that you would be taking those loans out regardless. For necessary medical care and health insurance that would cover you if you were hit by a car, your student budget should already include money designated for those expenses, and you can get a loan increase to cover medical expenses above the amount the initially offer. So it wouldn't necessarily be on credit cards.