I've now identified the "gaps" in my lineage for my DAR application. While two major gaps are in East Haddam, most are in Lyme. The next step for me, as always, is to make sure that I'm looking for early records at the right church. I've missed churches in the past because the actual church no longer exists - and thus, was missing from most histories of the town.
For Lyme, finding the right church has involved retracing the history of the town. Old Lyme (South Lyme) was initially part of Lyme, as was East Lyme. The area's oldest church appears to be the First Congregational Church of Old Lyme. According to the East Lyme Historical Society, it was established in 1665. Next, according to the Society, was the First Congregational Church of East Lyme in 1718, on the eastern side of Lyme. While I'm unsure of the date, I also should be looking at the records of the First Congregational Church of Lyme. Given my family's history,I probably should be looking at all three churches.