Adams, a Republican, says he will not "voluntarily commit a data breach" by handing over the documents to the Trump Justice Department.
Gov. Andy Beshear has also vowed to fight the attempts by the DOJ to obtain voter information, such as Social Security and driver's license numbers.
"This administration continually wants to get information on my citizens, Democrat, Republican and Independent, that they don't need, that they don't deserve, and that we promised that we would protect from private or public interests," Beshear told CNN earlier this month.
The governor added that he believes the goal is to suppress voter turnout by making some groups nervous about showing up at the polls.
The suit argues Attorney General Pam Bondi is uniquely charged by Congress with broad authority to request election records under the Civil Rights Act of 1960. She said in a statement the data is necessary to ensure "transparency, voter roll maintenance, and secure elections across the country."
Kentucky is far from alone. With the additional lawsuits, the Justice Department is now suing 29 states and the District of Columbia.