I am not familiar with Fulton County or Georgia’s specific procedures. But if the conduct is within the scope of their job, typically the governing body pays. If the conduct is outside the scope, the person pays. Or in the case of an elected official acting outside the scope, their campaign, PAC, or political party pays.
DAs and judges are sued all the time with outrageous, frequently frivolous claims of bias and misconduct (not commenting on this one at all—just saying as a general matter.) I had a case where the plaintiff went nuts and sued the federal judge for making a pass at her during the case; I was there, didn’t happen. So there is some good reason to provide them with representation. It often is the AGs office in the case of DAs. But certain states use private firms to keep fewer people on payroll.