I started out doing my usual search for "congregational church" + Putnam, hoping to find information on the town's oldest church. While I did turn up some basic history on the Congregational Church of Putnam, it didn't seem to date back to the 1700s... and then I remembered. Putnam wasn't founded until 1855. The residents would have attended the closest church in Killingly and other surrounding towns.
There's not much history on the Congregational Church of Putnam online. I suspect from the photos that it dates from the later half of the 1800s, right in line with the founding of the town. You can watch about their steeple restoration here. If you're interested in organ restoration, you can read more about their process here.
What I can find more about is St. Mary's Roman Catholic Church. There's a nice history on the town's website. With roots dating back to 1848, the first church was consecrated in the late 1850s. A new church, made mostly of brick, was consecrated in 1870. The church website provides more information. The second church was destroyed by fire and replaced in 1904. By 1956, the expanding parish renovated some church buildings and tore down others. The church is ethnically mixed: early pastors had French and Irish last names.
If you're looking for records, you can easily contact the parish using the information on their website.The cemetery is listed on Find A Grave.