As Sigma pointed out, if you apply for consolidation and REPAYE immediately after graduation, then you get start repayment right away, but also get the $0 monthly payments that you would not be able to get if you started repayment after starting intern year. That would mean you'd have no loan payments for all of intern year, and your interest rate would effectively be halved (the gov't pays half of unpaid interest if you're in REPAYE).
As for the credit card debt--you just need to pay that off ASAP. If you can't get a 0% balance transfer from another company (keep in mind there's usually a 5% fee, and that fee will collect interest), then you likely only have one other option for borrowing, which is to get a residency and relocation loan (Wells Fargo has them, or at least they used to offer them, and others like Sallie Mae, etc. offer them). The rate may not be much better than your CC's. I took out a residency and relocation loan during M4 and with a credit score in the high 700's my rate was ~9% if I recall correctly.
Also, if you haven't spoken with your financial aid counselor, you should. Depending on what your unforeseen expenses were, you may be eligible to ask for a budget/cost of attendance increase, which would let you borrow more in federal loans. Typically health-related stuff, car repairs, etc., are eligible. Not all schools will certify for an increase--mine did, but I don't think they do any longer.
Regardless, the best option is to pay that $20k on your credit cards as fast as possible. If you do get a 0% transfer or a residency/relocation loan, the goal of those is only to lower your interest rate--not your payment. You should still attack that debt ASAP.
You should know what your intern salary will be already (or at least know last year's intern's salaries), so go to paycheckcity.com and use their salary calculator to figure out what your take-home is. Then budget for the things you need like housing, utilities, food, etc. See how much you have left to work with, and hopefully you can pay off that $20k within the year if you're in a low cost of living area. Consider renting a room in a house, or sharing a 1bd. If you're single, a studio should be plenty big if you're not willing to share, but sharing is the best way to keep costs down. As an intern, you won't be home much anyway to enjoy your home--all you need is a safe and quiet place to sleep. So obviously avoid renting a room in a house with college students...
Assuming you consolidate/apply for REPAYE and get $0 payments for your first year, if you still have credit card debt after that year, and the rates are typical credit card rates (mid/high teens), then you're best off putting federal loans in deferment while you pay off that credit card debt. However, if you plan to take a job eligible for PSLF, then it may be worth holding on to the CC debt a little longer if that's what's required to keep making monthly loan payments.