Hello Ryan: I have been scanning census data for clues to the question of your Florence Bell Shaw's parentage. This has become a difficult issue as you no doubt know and I may simply be retracing your research steps in trying to fathom the issue. Here is a lead and perhaps it will ring a bell for you. If not then this lead can be forgotten.
Highland Township has as its principle city, Perrysville, Indiana. In the 1870 Federal Census (Ancestry.com)for Highland Township, Vermillion County, Indiana, p. 1, lines 1-6, and in the household of Asbury D. Beauchamp, age 39, born Indiana, and a minister, was a FLO BELL BEAUCHAMP, born in 1862 in Indiana.
Because of the similarities of the Flo Bell and your Florence Bell, including the year of birth as 1862, I wondered if there was an intervening marriage of your Florence to someone named Shaw. Or, barring that, perhaps Florence was later adopted. Of course, Asbury's wife's name was Nancy and NOT Hannah as you reported the name of your Florence's mother and this is an anomaly in this review. Also, Florence Lawlyes reported in census data that her father and mother were born in Indiana. Beauchamp was born in Indiana. These similarities prove nothing and are merely interesting leads.
I wish that I could do better. Based upon my experience as a family historian, I suspect that there was somehow, a family problem in the past of your Florence. What it may have been remains unknown. I say this because there was no discovery in Indiana of any Hannah (or Hanna) Shaw except one in the household of Hiram Shaw (1860 census for Highland, Vermillion Co., IN) and wife Rachel. This Hannah Shaw was 16 yrs. of age in 1860 and old enough by 1862 (18) to have born a child. I wondered, therefore, if perhaps your Florence may have been the result of a birth without benefit of married parents. I also looked at all Indiana marriage records and found none for a Hannah to a Shaw. It may be that a careful review of all documents in Vermillion County, Indiana, and also in Vermilion County, Illinois, might reveal an answer. I would suggest a review of land deeds, probate records, obituaries, and also a visit to the Illiana Historical and Genealogical Society in Danville, Illinois.
I do wish that I may have been more forthcoming. Good luck with your research.