Adcock ran strongly enough in the first primary to gain a runoff berth with the front-running former New Orleans city councilman James Edward "Jimmy" Fitzmorris, Jr. In fact, with 250,850 votes, he trailed Fitzmorris by just over 6,000 votes. Eliminated in the primary were two candidates from Webster Parish east of Shreveport, state Representative Parey Pershing Branton, Sr., of Shongaloo and Francis Edward Kennon, Jr., a Minden contractor and a nephew of former Governor Robert F. Kennon. But Fitzmorris was a convincing winner in the second Democratic primary and went on to win all sixty-four parishes in his race with the Republican candidate, former State Representative Morley A. Hudson of Shreveport, in the general election held on February 1, 1972.
Ironically, both Fitzmorris and Adcock later left the Democratic Party and became Republicans.
Adcock was also a past chairman of the Louisiana Tax Commission.
Adcock was survived by his wife, the former Frances Aycock (born 1921) of Monroe; a daughter and son-in-law, Jan Adcock Melton (born 1943) and Paul Anders Melton (born 1942) of Baton Rouge, and two grandsons, Jamar Adcock Melton (born 1970) and Paul A. Melton, Jr., (born 1972).
Adcock and a group of friends formed the Northminster Church in 1989. It is closely affiliated with the Alliance of Baptists, a fellowship dedicated to the preservation of historic Baptist principles and freedoms. He was a Sunday school teacher and a deacon.
Billy Boles, in a 2004 interview with Sam Hanna, Jr., of the weekly newspaper, the Ouachita Citizen in Monroe, recalled his friend Jamar Adcock: "I miss Jamar every day. He was just a great friend. There was not one thing that I could ask and he wouldn't do."
Adcock obituary, Monroe News Star, December 24, 1991
Report of the Louisiana Secretary of State, Democratic Primary Returns for Lieutenant Governor, November 1971