The LANSINGBURGH GAZETTE.—In 1781, the first number of the old Lansingburgh Gazette was issued by Messrs. Tracy & Bliss. For many miles on either side no other paper was then published. In Washington, Warren, Saratoga, Essex, Clinton and the adjoining counties of Vermont [Bennington, Rutland, Addison, Chittenden, and Grand Isle counties], the Gazette was almost the only paper circulated for several years.
We have now one of the first numbers before us. It is filled with sheriff’s and other legal advertisements from all the above named counties in this state. It is about twice the size of a foolscap sheet. Although the extent of the circulation has been curtailed, it has increased in size with the advancement in wealth, population and influence of the village, whose name it bears. It has been identified with the interests of the place—“growth with its growth, and strengthened with its strength.”Lans. Gaz.
Troy Daily Whig. September 9, 1839: 2 col 4.

In 1813, when all the New York state counties mentioned had been brought into existence (see e.g. “Interactive Map of New York County Formation History“), there were indeed not many local newspapers. There may be some errors below, but attempting to list those papers published in the 1810s and earlier should give some idea of how few – and how short-lived many of those few were. The bordering counties in Vermont had somewhat more, though many of those were also fairly short-lived.

In Washington County there was the Washington Patrol (1795-1795), the Northern Sentinel in Salem (1798-1804) which was to become the Northern Post (1804-1822), and the Washington Register (1803-1830) in Salem.

In Warren County there was the Warren Republican in Glens Falls was created that year and there wouldn’t be another in that county until 1818 with the Lake George Watchman.

In Saratoga County there was in Saratoga the Saratoga Register, or, Farmer’s Journal (1798-1800), the Republican Telescope (1801-1802), the Saratoga Advertiser (1804-1809), the Aurora Borealis; and, Saratoga Advertiser (1809-1810), and the Saratoga Courier (1815-1817?). In Ballston Spa there was the Independent American (1808-1818) which was to become the The People’s Watch-Tower (1818-1820), the Rural Visiter [sic] and Saratoga Advertiser (1812-1812) which was to become Saratoga Patriot (1812-1814), the Saratoga Journal (1814-1888), and the Saratoga Republican (1818-1823).

In Essex County there was the Reveille (1812-1816) Essex Patriot (1817-1819?) both in Elizabethtown.

In Clinton County there was the American Monitor (1809-1810) in Plattsburgh.