The ex had disappeared with kids and her parents refused to give the address. Luckily, she needed money and contacted my husband after about a year or so. He immediately started sending CS. The State of New Mexico came after my husband (for 10,000+) as his ex had claimed she wasn't receiving child support and got on state aid. He sent a copy of all our canceled checks, Western Union receipts, and money order receipts. They dropped collection efforts immediately. He called and asked if they were going to pursue fraud charges. The answer was no, unless she tried to collect aid in NM again.
Good luck. As long as the state has you to go after, they don't have to do anything else.