He attended the common schools in Bennington before attending Litchfield Law School in 1801. He studied law and was admitted to the bar in 1806.
He served as a United States Representative and United States Senator, and helped found the Whig Party. He began the practice of law in Bennigton before moving to Manchester to practice law. In 1809 he moved to Saint Albans to practice law.
He also engaged in banking and farming in the area.
He held various political positions in Vermont, and was elected to the Vermont State House in 1825. He served in the State House until 1827.
He was then elected to serve Vermont as a National Republican Party candidate in the United States House of Representatives. He served in the Twentieth and the Twenty-first Congresses from March 4, 1827 to March 3, 1831.
While in Congress, he was on the executive committee of the Congressional Temperance Society.
He declined renomination. In 1833 he was elected as an Anti-Jacksonian candidate to the United States Senate, serving from March 4, 1833 to March 3, 1839. Swift was not renominated for a second term in the Senate and returned to Saint Albans where he continued to work as a lawyer and farmer until his death.
He died on November 11, 1847 in Saint Albans, Vermont and is interred in Greenwood Cemetery in Saint Albans.
Member Vt; member United States House Member United States Senate from Vermont