YOu are correct. You need to find the local court where the first papers were filed before naturlaization took place.
You could order the films frm your local LDS. Here are the numbers:
New York County Court and Common Pleas
Declarations of Intention, 1802-46 FHL Film #0954904+
Declarations of Intention, 1846-95 FHL Film #0954072+
Declarations of Intention, all nations, 1889-90 FHL Film #0954903
Declarations of Intention, Germany, 1846-95 FHL Film #0947890+ (includes other countries)
Naturalization Indexes, 1853-66 FHL Film #0953637
New York County Courts
Index to Petitions, 1784-1906 FHL Film #0953636+
Naturalization Records & Index, 1792-1906 FHL Film #1001979+ This index includes the following courts:
New York County Court of Common Pleas, formerly Mayor's Court, 1792-1895
New York County Superior Court, 1828-95
New York County Supreme Court, 1868, 1896-1906
New York City Superior Court
Index to Naturalizations, 1843-1846 FHL Film #1953636+
Declarations of Intention, Great Britain and Germany, 1846-1895 FHL Film #0954630+
Original records of all of the above are at the New York County Courthouse in New York City.
New York County
New York County Clerk
60 Centre Street
New York, NY 10007-1402
Telephone: (212) 374-8361