Poking around for more East Haddam resources led me to a new source of record transcriptions. Jane Devlin, a genealogist from Michigan, has been digitizing transcriptions of church and cemetery records from the whole of Connecticut. The site's main focus is Middlesex County. Records are divided by town and then classed by record type. While, as always, you should choose actual images over transcriptions, this is a good starting point.
I didn't find the records I was looking for, but the site did offer me a few new hints. First of all, there are far more Graves listed in the Millington church records than in the East Haddam Congregational Church. The village is closer to Lyme than the East Haddam Church (which is close to the river), so this was probably my ancestor's church. The building was destroyed almost forty years ago. I had never heard of it and never would have thought to look there. Second, the Barbour Collection is less complete than I had thought. After 1800, it only includes marriages. Good to know. Hopefully I'll be able to find what I need.