Will going to court help us get out of paying welfare reimbursement to husbands ex wife?

My husband's ex wife quit her job in October. Her and my husband share joint custody of their 13 year old daughter. They also have a son who just turned 18 who has lived with us full time for 3 years and we have supported him financially 100% for at least five. The ex wife filed for welfare in January and was granted it. At the time, she was living by herself with no source of income. Since then, she has moved in with her boyfriend, and the father of her unborn child. She is still collecting welfare and now they want us to pay 485 a month. Will we have to pay if we fight it? We have his daughter half the time, and actually support her financially more than her mom. Not only do we pay for her health and dental insurance, but her mom never buys her new clothes and we've paid for all the school supplies for at least two years. She willingly quit a job and turned down another job offer. I'm wondering if it's possible for us to win this. I could see if her mom had her full time and us just on the weekends them coming after us for support. But she's here 50% of the time, sometimes more and is fully supported. Plus, the mom's boyfriend works. They live in apartments that are over 800 dollars a month, they have internet, cable and all that. Why should she be able to collect welfare? I think it's fraud if someone else is paying the rent and bills. Not to mention she's pregnant with his child..what happens after it's born? Shouldn't he pay? The whole thing just irritates me. We're in the process of remodeling our home, and starting fertility treatments so that we can have a child as I suffer from endometriosis and have been in bad health. Over the past few years we've been able to pay off debt and are finally on the road to good things and I can't fathom having to pay her 485 a month. Because she willingly quit a good paying job. Do you think the court would side with us and we won't have to pay?