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I have gone through all of my records, but I have no information indicating a second Braxton Dunleavy. However, there certainly could have been another, perhaps a child of William, James, or John H,Dunleavy who were Braxton's brothers. If so, it doesn't appear that they lived in Middlesex County.
I would suggest that you check census records to see if you can locate your Braxton. The 1797 time frame suggests this this second Braxton is most likely a part of this family. I wasn't searching for Dunleavys other than this Braxton and his Father, John. Our Braxton had two daughters: Mary & Nancy, and no mention of any others.
I do have the marriage records from 1768 to 1849 in middlesex County, but there is only the Braxton who married in 1797.