Maybe it would be useful to list the basic facts about this story. It's a short one; the Silverstone book Schoonerbum mentioned covers it in less than four pages - including pictures. The following information is condensed from that book. The parenthetical weblinks are to the relevant entries in the online version of the Dictionary of American Naval Fighting Ships. (I don't have a copy of Don Canney's book, but I suspect he doesn't have much more to say about American ships of the line - for the simple reason that there isn't a lot more to be said.)
If I've counted correctly, a grand total of ten sailing vessels that fit a reasonable definition of the term "ship of the line" flew the flag of the U.S. Navy - and three of those fit the definition only if it's stretched considerably.
In 1799, during the brief dominance of the Federalist party (and the Quasi-War with France), Congress authorized the construction of six ships of the line. Only one had been laid down when, in the following year, the conflict with France cooled down and Congress canceled the program. None of those ships was ever launched.
Two ships of the line, named Independence and Washington, were authorized, laid down, and launched during the War of 1812, but neither was commissioned before hostilities ceased. Both had bad reputations in terms of sailing qualities. The Independence ( http://www.history.navy.mil/danfs/i1/independence-ii.htm ) served as flagship of the Mediterranean and Home Squadrons for a few years before being laid up "in ordinary" from 1822 to 1836. She was then "razeed" (i.e., converted more-or-less into a frigate by the removal of her upper gundeck), and had an active career on various stations around the world until she was assigned as a receiving ship (i.e., a floating barracks for newly-recruited sailors) in 1857. She was decommissioned in 1912 and broken up the following year. The Washington ( http://www.history.navy.mil/danfs/w3/washington-iv.htm ) served for two years as flagship of the Mediterranean Squadron, went into reserve in 1820, and was broken up in 1845.
Two more ships of the line, the Franklin and Columbus, were authorized during the War of 1812 but not laid down until 1815, after the Treaty of Ghent. The Franklin, like the Independence and Washington, was found to be a clumsy sailer, but saw active service in the Mediterranean and the Pacific until 1824 ( http://www.history.navy.mil/danfs/f4/franklin-iii.htm ). She was then laid up until 1843; in that year she was put back into service as a receiving ship. She was finally decommissioned in 1843 and broken up ten years later. (Some of her timbers may have made their way into a new steam-powered screw frigate of the same name, which was built with funds authorized for "rebuilding" the old ship. It used to be thought that this process, which also was used in the breaking up of the 1797 frigate Constellation and the construction of a new corvette of the same name, was a form of deception perpetrated by the Navy on the Congress. Recent research has established that it was, in fact, perfectly legal, and almost certainly understood for what it was by everybody concerned.) The Columbus (http://www.history.navy.mil/danfs/c11/columbus-ii.htm ), though Silverstone describes her as being known as a "poor sea boat," had quite an active career until she was laid up in 1848. She was burned to avoid capture at the Norfolk Navy Yard in 1861 (along with three other ships of the line; see below).
Right after the War of 1812 the Congress went on a modest military spending spree, which included the authorization, on April 29, 1816, of nine ships "to rate not less than 74 guns each." Seven of these actually got built, but it took a while - and only three of them actually did the sort of service for which they had been designed.
Six of the authorized nine were sister ships of 74 guns each. Two of them ever served in their intended roles:
Delaware ( http://www.history.navy.mil/danfs/d3/delaware-iii.htm ). Laid down 1817, launched 1820, commissioned 1828. She had a more-or-less active career until 1844, when she was laid up at Norfolk; she was burned when the Navy Yard there was evacuated in 1861.
North Carolina ( http://www.history.navy.mil/danfs/n6/north_carolina-i.htm ). Laid down 1816, launched 1820, commissioned 1825. Was in active service intermittently until 1840, when she became a receiving ship. Sold in 1867.
Two other members of the class did get launched and commissioned, but never served their intended purposes:
Alabama. Laid down 1819; launched 1864 (after what surely must be some sort of record for time under construction). In 1863, for obvious reasons, she had been renamed New Hampshire. ( http://www.history.navy.mil/danfs/n4/new_hampshire-i.htm ). She was launched as a storeship, renamed Granite State in 1904, and sold in 1921.
Vermont (ex-Virginia, ex-Massachusetts) ( http://www.history.navy.mil/danfs/v2/vermont-i.htm ). Laid down 1818, launched 1848, commissioned 1862. (Silverstone notes that she was "ready for launching in 1825 but only launched to clear slip in 1848.") She was commissioned as a storeship for one of the Union blockading squadrons. She was sold in 1902.
The other two were never launched:
New York. Laid down in 1820 at Norfolk. She was still under construction at the start of the Civil War, and shared the fate of the Columbus and Delaware.
Virginia (ex-Vermont). Laid down 1822. Silverstone says she was "ready for launching, 1825 but never launched. B[roken] U[p] on stocks, 1874."
Silverstone, perhaps a bit euphemistically, describes this class as "highly successful vessels. [Well, maybe he's just talking about the Delaware and North Carolina.] Alabama, New York, Vermont, and Virginia were completed ready to launch and kept in reserve until needed...." Those must be the ones Chuck Fan mentioned - though they were authorized after, not during, the War of 1812. (None of them was still under construction in the 1880s; all of them had been either launched or burned on the stocks by the end of the Civil War. Only four - the razeed Indepencence, the Alabama/New Hampshire, the Virginia/Massachusetts/Vermont, and the Ohio - were still Navy property as of 1880. I haven't encountered that story of a ship of the line being destroyed when a temporary barn fell down. I suppose it could have happened to one of them after being decommissioned and sold.)
Two of the other ships of the line authorized right after the War of 1812 were one-of-a-kind:
Ohio, 74 ( http://www.history.navy.mil/danfs/o2/ohio-ii.htm ). Laid down 1817, launched 1820, commissioned 1838. She was considered a good sailer, and unusually fast for her size. (The DANFS says she was reputed to have reached 12 knots.) She was sold in 1883.
Pennsylvania, 120 ( http://www.history.navy.mil/danfs/p4/pennsylvania-i.htm ). The largest sailing warship ever built for the U.S. Navy, she had four full-length gundecks. (Her designer, Samuel Humphreys, apparently consulted the plans of the famous Spanish ship of the line Santissima Trinidad; there's a copy of the latter vessel's hull lines in the U.S. National Archives.) She saw no active service; she was made into a receiving ship at Norfolk shortly after she was commissioned, and was burned to prevent capture by the Confederates at the start of the Civil War. That fire consumed four of the eleven ships of the line the U.S. Navy ever built - along with the steam frigate Merrimack, which had a brief but interesting second career thereafter.
Ship number nine of the 1816 authorization apparently died somewhere in the bureaucratic process.
So only seven American ships - the Independence, Washington, Franklin, Columbus, Delaware, North Carolina, and Ohio - ever did active service as ships of the line. Three others - the Alabama/New Hampshire, Virginia/Massachusetts/Vermont, and Pennsylvania - were authorized and laid down as ships of the line, and were launched and placed in commission, but were never used for their original purpose. Two more, the New York and Virginia, were laid down and almost finished as ships of the line, but were destroyed before they were launched.
The big factor that led to so many American sailing warships being under construction for so long was money. Federal budgets in the nineteenth century were, by modern standards, incredibly small. (There was no federal income tax in those days; most of the money had to come from import tariffs, excise taxes, and the sale of government bonds.) The Navy (and the Army too, for that matter) got a low priority. These were the days of the Monroe Doctrine, when the United States professed to have no military or naval interests beyond the western hemisphere. For that matter, callling these vessels "ships of the line" could be said to be a slight misuse of the language; the U.S. Navy was never set up to form a line of battle in the traditional European sense.
Over the years those ships have attracted a surprising amount of attention from model builders. If I remember right, the Smithsonian used to have a nice model of one of them in its naval/military history gallery. (Almost all the Smithsonian's warship models got left out of the new military gallery that opened a few years ago; that's one of my few criticisms of it.) I've seen several other nice models of American ships of the line in various places. Most recently, the visitor center on the pier beside the U.S.S. North Carolina, in Wilmington, N.C., commissioned a series of excellent models representing all the warships named North Carolina, including the old ship of the line. Anybody wanting to see that one would be well advised to do so ASAP. Sometime in the fairly near future the great battleship herself is slated to get towed up to Newport News for some major - and long overdue - repairs to her hull.