He was admitted to the bar in 1818 and commenced practice in Wytheville, Virginia.
Draper was elected as a Jacksonian to the Twenty-first Congress to fill the vacancy caused by the death of Alexander Smyth and served from December 6, 1830, to March 3, 1831.
He unsuccessfully contested the election of Charles C. Johnston to the Twenty-second Congress.
Draper was subsequently elected to the Twenty-second Congress to fill the vacancy caused by the death of Charles C. Johnston and served from December 6, 1832, to March 3, 1833.
He resumed the practice of law until his death in Wytheville, Virginia, June 10, 1834.