Can a co signer sue the person he/she signed for?

I need to do elective surgery for my nose because I got into a car accident. So my nose is broken and I need someone to cosign for me because I do not have credit. I'm only 18 and I was asking everybody in my family if they could co sign for me. But they don't think that I will pay each month. So I just want to know if they could sue me because obviously I don't want to be sued, so I will pay my bill.