Drowne's father (also named Solomon, as was his father) was a merchant and was heavily involved in the civic affairs of Providence and active in the First Baptist Church in America. In 1772, Drown witnessed the burning of a British ship in an event known as the Gaspée Affair. The following year, he graduated from Brown University and continued his academic career, receiving medical degrees from Dartmouth College and the College of Philadelphia.From 1776 to 1780, during the Revolutionary War, Drowne served as a surgeon in the Continental Army. On July 3, 1776, he narrowly escaped being captured by the British while gathering medical supplies in New York City. While stationed and treating soldiers at Rhode Island, he gained the favor of Marquis de Lafayette, the Comte de Rochambeau, the Comte d'Estaing and other French military officers supporting the American cause. In autumn 1780, he became surgeon on board the privateer sloop Hope owned by Joseph Nightingale and John Innis Clarke, keeping a diary that was later published. Drowne was discharged from the First Rhode Island Regiment on June 15, 1783, receiving a Badge of Merit for six years' faithful service.
In 1783, Drowne was elected to the Brown University board of fellows. He traveled to Europe, touring various medical facilities and schools and meeting Benjamin Franklin and Thomas Jefferson in Paris. Returning home, he practiced medicine in Providence, Rhode Island until 1788, when he settled in Marietta, Ohio with other war veterans.
As his health declined, Drowne moved from Marietta to Morgantown, West Virginia and Union, Pennsylvania. In 1801 Drowne returned to Rhode Island and bought a farm next to Senator Foster in Foster, Rhode Island named Mt. Hygeia after the Greek goddess of health. Drowne used the farm for botanical research and named his driveway the "Appian Way". In 1811, Drowne was appointed Professor of Botany and Materia Medica at Rhode Island College. He laid out the college's first botanical garden, became one of the original members of the Rhode Island Medical Society and American Academy of Arts and Sciences; and one of the founders of the Rhode Island Society for the Encouragement of Domestic Industry. With his son William Drowne, he co-wrote The Farmer's Guide, a thorough guide on husbandry and gardening, in 1824. Documented service in the DAR records:
DROWN, SOLOMON (Staff Officer) DAR Ancestor
Service: RHODE ISLAND Rank(s): STAFF OFFICER
Birth: 3-11-1753 PROVIDENCE PROVIDENCE CO RHODE ISLAND
Death: 2-5-1834 FOSTER PROVIDENCE CO RHODE ISLAND
Pension Number: S*W8201
Service Source: S*W8201
Service Description: 1) SURGEON & SURGEON'S MATE, IN THE CONTINENTAL ARMY UNDER COL CAREY
RESIDENCE 1) City: PROVIDENCE - County: PROVIDENCE CO - State: RHODE ISLAND
SPOUSE 1) Elizabeth Russell

In addition the following children are also documented in the DAR:
Henry Bernadin, Julia Ann Stafford,
William, Emily Day
Solomon Horace, Susan Leonard