Sometimes the churches predate the formation of the town... And that's true of Salem, according to the church's website.
While the town of Salem was founded in 1819, the Ecclesiastical Society of New Salem was actually founded in 1725. The Society included residents of Lyme and Colchester. The current church appears to date by 1838. Since 1890, the church has existed independently.
I'm unclear as to whether the church records are included in the State Library's slip index. You can always contact the church for more information.