You have some options. If you put it onto a credit card, you would only have to make the monthly payments to keep it in line. If you are struggling, you could contact that credit card company and they would very likely work out a lower payment plan for a period of time if necessary. They would rather do this than lose the money in a bankruptcy.
I would really try to work it out if you can. If not, you would most likely use bankruptcy as a way to pay down your debt. Very few people are eligible for chapter 7 bankruptcy any longer and you would most likely fall into chapter 11 bankruptcy. In this case, your amount owed to each lender would be lowered and monthly payments be made to pay down your debt at a lower amount due.
It is more beneficial to try to work things out with your lenders as declaring bankruptcy will really have a negative effect on your credit rating for many years into the future.
First and foremost, you might want to contact a bankruptcy attorney (you will need one anyway) - many of them will give a free first session to discuss the issue.
Good luck as you try to work out your issues - it is highly understandable in today's economy. Hope all works out as well for your dental work.