So far, based on what was written here, I think there are 6 basic methods:
1 - get credit card counseling. This option can be done together with any of the others.
2 - keep paying minimum payments and try to pay off one card at a time.
3 - borrow money from a gemach or financial institution to pay off credit cards, and pay back the new consolidated loan with lower interest, or interest-free. Downside about this is that it's hard to be approved for a regular loan without a good credit score, and it's a hassle (I would assume) to get a gemach loan. (Also, a gemach would probably mean having others involved, and this is obviously a very private matter.)
4 - default on the loans just long enough to be able to negotiate a really good deal, which usually would not be available if the loan was in good standing. The deal could be one of two things, either pay off a reduced loan amount upfront, or pay off the entire loan over time with a reduced interest rate.
5 - enter a debt repayment plan through a third party. Basically the same as #3, but having it done by others. They do the work, but costs commission.
6 - declare bankruptcy. This obviously impacts credit score the most and impacts assets already owned (IINM).