I am looking for a birth record for my 2G Grandfather, Augustus Willard Gilman. Census records say he was born in Middletown, CT, but I am unable to find a record of his birth on Ancestry or FirstSearch. His parents John (13 Mar 1814 - 2 Sep 1890) and Polly Doolittle (10 Oct 1812 - 28 Dec 1890) were married in Shrewsbury, VT on 21 Oct 1838 as per VR. This date is five years after Augustus' birth, so I am wondering if there might have been an original parent that died or if his birth either wasn't recorded for some reason or is in an archive outside of Ancestry and FamilySearch. I have already checked the Barbour collection in Ancestry and have come up empty.
Does anyone have any ideas about where else I may look or have an understanding of birth recording and marriage practices in 1830s Connecticut/New England?
I have attached the 1870 US census for Middletown, VT that states Augustus was born in "Conn". There may be a possibility that there is some confusion between Middletown, CT and Middletown, VT.
Thank you in advance.