Abiezer Veazie was born on August 20, 1831, in Rockland, Maine, to Abiezer and Grace Veazie. His father died on May 11, 1839, and his mother followed around 1851. By 1850, Veazie was living in Camden, Maine, and working as a laborer. He married Freelove Richards on April 19, 1851, and their daughter Freelove Adella was born around 1852. By 1860, Veazie was working as a teamster in Rockland, Maine, and he owned $600 of real estate and $300 of personal property.
Veazie enlisted in the Union army on May 17, 1861, and mustered in as a private in the 4th Maine Infantry. He mustered out on September 16, 1861. He reenlisted on February 1, 1862, and mustered in as a private in Company B of the 1st Maine Cavalry. The regiment took part in the Second Battle of Bull Run, the Battle of Antietam, the Battle of Fredericksburg, the Battle of Gettysburg, and the Battle of Cold Harbor. He was assigned to duty as a teamster toward the end of the war, and he mustered out on August 14, 1865.
Veazie returned to Rockland after the war. By 1870, he was working as a “Truckman,” and he owned $1,500 of real estate and $400 of personal property. He helped care for his elderly mother-in-law, Margaret Richards, as well as several other relatives. He applied for a federal pension in May 1879 and secured one soon afterward. He died on August 14, 1890.