Credit counseling and debt management courses are really only of use if you've got real income. You don't. So you're going to need to try something else.
Dropping out of school to work full-time is not a realistic option. If you really do have $15,000-$20,000 in credit card debt then you're in pretty big trouble. At an average credit card interest rate, even just paying the minimum would take about $300 a month for 20 years and cost north of $70,000 dollars by the time you've paid it all off. As any job you're likely to be able to get without a college degree isn't going to pay much more than $20,000 a year, you'd wind up having to spend $3600 a year just on debt management, and you'd have to do that for 20 years. You can't afford that. No one can.
Bankruptcy might be an option, but not a particularly attractive one should you want to keep going to school. I assume that you've got student loans and will need more to stay in school. Lenders will be extremely reluctant to loan money to anyone who as declared bankruptcy, especially someone as young as you are. So while that could conceivably eliminate a lot of your debt, it would almost certainly permenantly damage your credit.
It's also not a given that you could even declare bankruptcty. The rules for doing so were recently tightened by Congress. I don't know the exact details of your situation or if you'd be affected, but it's harder than it used to be.
As unpleasant as it sounds, I believe that declaring fraudulant use of your cards may be your best and, in fact, only option. You yourself did not run up the charges on the cards. Your family used the cards without your knowledge, knowing they were screwing you over, and are now unwilling to help you get out of the mess they put you in. Any loyalty to the person who did this to you is entirely misplaced. While it's true that you don't generally want to level fraud charges against your family, it's also true that you don't generally want to defraud your family.
You need to get in touch with your credit card companies and tell them that your cards have been used fraudulently. If you can tell them who did it, they'll go after the fraudster themselves, more than likely. You may need to get in touch with a lawyer to prove that it really wasn't you who ran up these debts. Either way, this sounds like the only way that you're going to ever be able to get out of this mess.