I suspect they would have control of the account that the state set up to distribute benefits since they had enough information to file the claim in the first place; but the non-mailing contact info would be email and phone number that they control. Once the funds have been distributed, they'd be notified and then make the transfer to the second account set up at another bank. From there...poof...

This is probably why a second bank account was opened up in my name. This bank's KYC controls were so lax that they didn't even bother to do a credit inquiry on me, which would have resulted in a notice to me from a credit monitoring service that I use.