Mr. Jones said Mr. Martin's "poor decisions and his bad management" as head of the state's Department of Human Resources contributed to the death of children in state care. Mr. Martin again sought to link Mr. Jones -- who voted twice for George W. Bush -- "to the failed policies of the Republicans."
"The main difference between me and Vernon Jones is I'm a true Democrat," Mr. Martin said.
The two men have criticized each other's presidential picks since the July 15 primary.
Mr. Martin voted for John Edwards in Georgia's Feb. 5 primary even though the North Carolina senator had already dropped out of the race. Mr. Jones said Tuesday that Mr. Martin had "voted against" Barack Obama, who coasted to a win in Georgia with overwhelming support from the state's large black community.
"He did not think Barack Obama should be president," Mr. Jones said. "He voted for someone who was not even running for president."
Mr. Martin bristled at Mr. Jones' suggestion that he was to blame for the deaths of children in state custody.
"Vernon knows that's not true. Vernon knows my life's commitment has been to children," Mr. Martin said.
Mr. Jones said he wasn't questioning Mr. Martin's commitment to children but his management skills. He also blasted Mr. Martin for cutting DHR funding for programs that help children. Mr. Martin said such cuts were necessary when the economy sputtered after the Sept. 11 terrorist attacks.
Mr. Jones, the chief executive officer of DeKalb County, faced questions about his temperament, citing a 2004 confrontation with a Republican county commissioner who accused Mr. Jones of shoving her after a heated budget dispute. Mr. Jones wrote a letter of apology at the time. On Sunday, he said his personality brought about positive changes in DeKalb County.
"I'll say this, though: It was my temperament that responded to those (Hurricane) Katrina victims, it was my temperament that created 3,000 acres of green space, it was my temperament that created jobs," he said.
The winner of Tuesday's runoff will face Republican incumbent Saxby Chambliss and Libertarian Allen Buckley in November.