Adams Island is alongside Middleburgh, a few blocks south of what had once been (prior to Batestown being annexed by the City of Troy) the southernmost boundary of the Town of Lansingburgh at Ingalls Avenue. (Incidentally, Van Schaick Island was for a time also called Adams Island.)

AN application will be made by the subscriber and his associates to the legislature of this state at their next session, for a grant of the Island or sand bar in Hudson’s River, immediately below the dam between Troy and Lansingburgh. dec 12, 1825. 60F6
JOSEPH FRIOT.
Albany Argus & Albany Gazette. February 3, 1826: 4 col 1. [Adams Island or some other?]

In SENATE.—Feb. 7.
Petitions—for the incorporation of the Lehigh coal company; of Jacob Friot, and others, to be allowed to purchase an island in the Hudson river, between Troy and Lansingburgh.
Geneva Gazette and General Advertiser. February 15, 1826: 2 col 5. [Adams Island or some other?]

LEGISLATURE OF NEW-YORK.
[Reported for the Albany Argus & Daily City Gazette.]
IN SENATE,

Saturday, Feb. 18. […]
A communication was received from the commissioners of the land office, with an unfavorable opinion upon the petition of Joseph Friot and others, for the purchase of an island in the Hudson river.
Albany Argus & Daily City Gazette. February 20, 1826: 2 col 5.

LEGISLATURE OF NEW-YORK.
[Reported for the Albany Argus and Daily City Gazette.]
IN SENATE,

Tuesday, April 10. […]
The remonstrance of Van Der Cook against granting to Elijah Adams a certain sand bar in the Hudson, was presented.
Albany Argus & Daily City Gazette. April 11, 1827: 2 col 4.

CHAP. 311.

AN ACT authorising the Commissioners of the Land-Office to convey to Elijah Adams a Sand Bar in the North River, between the City of Troy and the Village of Lansingburgh.
Passed April 16, 1827.
1. BE it enacted by the People of the State of New-York, represented in Senate and Assembly, That it shall be the duty of the commissioners of the land-office to quit claim to Elijah Adams all that island or sand-bar, that rises above the water, situate in the North river, between the city of Troy and the village of Lansingburgh, and lying westerly of the place where the island of Irondequot, or Adams island was situate, containing about six acres of land.
2. And be it further enacted, That before the said commissioners of the land-office shall be authorised to quit claim the said island or sand-bar as aforesaid, the canal appraisers shall, by the best means in their power, ascertain the value thereof, having reference in their appraisement to all fishing privileges appurtenant thereto; and it shall be the further duty of the said canal appraisers to ascertain the value of such part of Irondequot or Adams island, before the erection of the dam near the same, as in their opinion may have been carried away by reason of the erection of said dam; and if the value of the first mentioned island or sand-bar shall exceed in value the said Adams island, the said Elijah Adams shall pay the amount of such excess to the treasurer of the state; and the conveyance of the said sand-bar to the said Adams shall be taken to be in full satisfaction for all damages which he may have sustained by reason of the erection of the said dam.
Laws of the State of New-York at the Fiftieth Session of the Legislature. Albany, NY: E. Croswell, 1827. 348.

Detail of Islands south of the Dam and Sloop Lock cropped and edited from
Rogerson, A. E, E. A Balch, and Robert Pearsall Smith. Map of Rensselaer County, New York: from actual surveys. [Troy N.Y.: E.A. Balch, publisher, 1854] Map. Retrieved from the Library of Congress, .

Detail of Islands south of the Dam and Sloop Lock cropped and edited from
Lake, D. J, and S. N Beers. Map of Rensselaer Co., New York. Philadelphia: Smith, Gallup & Co. Publishers, 1861. Map. Retrieved from the Library of Congress, .