I've found a bit more to add to this information, which seems to add proof that we're on the right track.
I have found a record of a Sarah Estabrooks, born 10 October 1764 (in Cornwallis, Nova Scotia, Canada), died 2 January 1844 (in Fredericton, York, New Brunswick, Canada). This Sarah was the daughter of Elijah Estabrooks Jr and his wife Mary Hackett. Sarah was buried in the Loyalist Cemetery in Fredericton three days after her death in 1844, due to the fact that her husband, JOHN MARSH, was a Loyalist who received a grant of land in Canada. JOHN MARSH and SARAH ESTABROOKS were married in Canning, Carleton County, New Brunswick on 15 July, 1790.
I am convinced that this is MY John Lathrop Marsh (Sr), son of Simeon Marsh and (probably) Eunice Lathrop of Connecticut.
If so, I know that John and Sarah Estabrooks Marsh had at least one child: a son, named John Lathrop Marsh Jr. (I believe there were several other children too but unfortunately I do not have their names. John L. Marsh Jr. was probably one of the oldest, if not the first child born to John Sr and Sarah.)