A legal name change is the obvious solution, here. A thorough background check will certainly turn up the legal name change and the sex offender status, but few employers actually perform a thorough background check.
For the most part, you only need to pass the "what comes back in a Google search for your current legal name" test.
Yes, this might limit OP's options when it comes to prospective employers, but there are plenty whose "background check" is "have someone in HR type their legal name from their government-issued ID into Google".
Of course, this all _takes time_ which OP doesn't have, but if they hadn't left this to the last minute, this wouldn't have been an issue.
"The best time to plant a tree is 20 years ago. The second best time is today."
