Not sure if you have seen this from Copinger - The Manors of Suffolk, vol. 4
https://archive.org/stream/cu31924092579576#page/n77/mode/2up
This (paragraph 3) gives his parents as another Stephen Baxter and his wife Cecilia, and his heirs as 4 daughters, so he can't have had a son who survived him as is currently on Wikitree (Baxter-3409). It doesn't say who was the mother of the 4 daughters but I doubt they are all the children of Elizabeth Freston.
I can't see any Baxters in the Visitiation of Sussex, but there are some in the Visitation of Norfolk, that could be related. One in particular uses the first name of Stephen a bit. https://archive.org/stream/publicationsofha32harluoft#page/22/mode/2up
FreeREG has the burial of Stephen Baxter at All Saints, Mendham on 14 Sep 1696 and also burial of Alice the wife of Mr Stephen Baxter on 05 Sep 1693 (perhaps a second wife) and burial of Mary Baxter on 26 Dec 1679 daughter of Stephen Baxter, Gent (perhaps a daughter or sister?). Unfortunately the Parish records only start in 1678.
I think you might have already had a link to Francis Blomefield. "Hundred of Earsham: Mendham," in An Essay Towards A Topographical History of the County of Norfolk: Volume 5, (London: W Miller, 1806), 372-387. British History Online, accessed March 25, 2016, http://www.british-history.ac.uk/topographical-hist-norfolk/vol5/pp372-387. which has this in Mendham Church "Depositum Stephani Baxter Generosi, qui decessit 12 Die Sept. 1696, æt. 79,"
