While you are on First Offender probation, there is no conviction. If you successfully complete the probation, there will never be a conviction entered. Therefore, you should still be eligible unless and until you get the First Offender status revoked.
When you take a First Offender plea, most judges will say something like "You can honestly tell employers (and GA Dept of Human Services) that you've never been convicted of this crime". Do well on probation and you should be fine.
Once you successfully complete probation and receive a first offender discharge you will not have a conviction on your record in Georgia and the case should not effect your benefit eligibility.