You are wise to refuse to co-sign the loan. I truly hope they are not pressuring you, or getting all emotional over it or threatening that they will never speak to you again, or anything like that. I hope they can just say, "OK, we understand" even if they don't.
Meanwhile: Do you have a place that's your regular bank? You can ask any time to meet with a personal banker face to face. Use that and get financial counseling for yourselves (because combined income of $30,00 a year is frankly very low, and you need to be sure your own money is well handled) and then talk with the bank about where your relatives can go to get advice. They may need to be put on to payment plans with their creditors (and most utility companies, phone companies, etc. will indeed help people get onto payment plans to pay down debts to those kinds of firms). If they are about to be foreclosed upon -- have they talked to whoever holds their mortgage about a new payment plan? There are also government agencies and programs to help families facing foreclosure. Help them find these programs immediately, help them deal today with the utility company, the phone company, etc.
Your own banker should be able to start pointing you to this help for your relatives. Ask the bank where your relatives can go to get serious financial advice, get help managing down their debt and somehow get help period--they are clearly not living "beyond their means" or spending frivolously but they are nearly homeless! So they need immediate and professional help.
Professional help -- not more little loans and certainly NOT your signature on a loan. You sign, you're liable. Your family of three has only $30K a year which in our area would be on the verge of poverty, honestly, so don't tangle your finances with theirs in any way.
I really hope you can get them some help and that they will work hard and fast to get advice, sign on to programs and tell their creditors they need payment plans immediately. It is tragic to see people who are living very frugally but still having trouble, and I hope they can be assertive about getting help and aren't afraid to ask for the help that's out there.