In most states, unless you agree to pay more, you are responsible for only 6 years of back support, I believe. Look this up. If this is true, don’t sign anything. It would be in the laws of the state where your ex lives.(family law) Also, if she was on welfare at anytime that you were paying her support, she would be in trouble for not reporting it to the state.
As a rule they have to collect from you not her. If she was collecting from you, she wasn’t reporting it. You should be able to get the number of the fraud unit on the state human services page. In my state, they only keep welfare records three years, unless fraud is suspected. If you call they will ask you questions about whether she is cohabitating, what kind of car does she drive, is she living with anyone, where do they work, things like that.