Doing a bit more thinking about Elizabeth's life, I suspect we will find very little about her in the records of St. Mary's in Dodgeville (stored in West Burlington) but I do have someone there taking a look for me. I think it's more likely we'll find information about her early life at St. John's in Burlington. I have an appointment set up to view their original church books in about a week and a half. You know I'll be enjoying that visit! While the trip is still not set in stone, the current plan is to bring my mother to visit her sisters October 3-6. I plan on visiting St. John's to view their records on that Friday. You know I'll let everyone know what I find.
Lesson: Even if you think you've looked everywhere for the elusive ancestor's records, re-read everything you have on them. Really think about where you'll find something about them. All this time, I've been focusing on Dodgeville's records when in fact, we should be able to find her in Burlington's records. Close to each other, sure, but they are not the same jurisdiction. I'm hopeful I'll find a little more about her at St. John's Church in Burlington, then, hopefully, find her birth record on the correct date in New York City.