I ran some Goodman names through Family Search’s full text search and was able to answer one or two important questions! But at the same time, another discovery cast more doubt on what was already a shaky branch on the tree.
I knew for sure that my 3g grandfather was Timothy Goodman (1825-1906). My 2g grandmother, Lucy Ann Goodman, is clearly found in his household in 1860 and 1870. So no problem there.

While Timothy is always found on censuses and other residence documents as “Timothy” or “T” Goodman, he is referred to on many family trees as “John Timothy Goodman” with no sources to substantiate this fact. While it is true that my Timothy had a son named John Timothy, that was not enough proof for me. I decided to dig into court records to see what might come up.
I found enough evidence to match my Timothy Goodman to a “John T Goodman” who resided in Dooly County at the same time. The first was an 1849 probate document naming John T Goodman as a legatee of Timothy Goodman. The older Timothy had already been established as the father of the younger Timothy. Furthermore, no John Goodman appears in local records at the time, and there is no other known child of Timothy I who could be this John. So, it certainly appears that John T Goodman is the same person as the younger Timothy Goodman.

Secondly, this John T Goodman purchased 75 acres of public land in Hamilton County, Florida in 1856, while still residing in Dooly County. Many years later, in 1895, Timothy Goodman claimed what appears to the same tract under the Homestead Act. He then sold off the land in two tracts later that year. This matches up John T and Timothy a second time, and also explains how John Timothy ended up in Hamilton County in his latter years, arriving around 1885. He apparently moved to live on the land that he had bought as an investment three decades earlier.
So, that settles one question that had been hanging out there. My 3g grandfather Timothy Goodman was indeed John Timothy Goodman.
I was able to discard another theory. Many trees have this John Timothy Goodman as married to a Rebecca Pursell. His first wife was indeed named Rebecca. The surname Pursell comes from a marriage record to a John Goodman in 1849 in Troup County, Georgia. However, there was a John S Goodman living in Troup County at that time. Troup County is also nowhere near Dooly County. So, I do not think that this is not the same John, nor is it the same Rebecca. My Rebecca’s surname remains unknown.
I then moved on to John Timothy’s father, a Timothy Goodman who first lived in Wilkinson, then in Lee County, possibly winding up in Dooly County shortly before his death around 1849. While this man left hardly any paper trail at all, I do think this is the correct Timothy, and the father of John Timothy, based on some ties between John Timothy Goodman and both Wilkinson and Lee counties, as well as the probate record mentioned above.
Many online trees have this earlier Timothy as the son of yet another Timothy from Dobbs County, North Carolina. I have not seen any proof of this: it just seems to be a widely held assumption. I ran across a deed, however, that makes me question this connection. In 1834, Timothy Goodman sold land in Stewart County, Georgia (in a part that was formerly Lee County) as the administrator of Joel Goodman, “an orphan of Plummer’s District, Lawrence County.” A further search showed that this land was granted in 1827 to “Joel M Goodman.”
“Lawrence County” is actually Laurens County, SC, which had a Plummer’s District. And a Timothy Goodman is found in Laurens County! So, what was my Timothy’s connection to Joel? An “orphan” would normally mean a minor child, so my guess is that this was a younger brother or a nephew? Certainly a relative of some kind.
In what is probably not a coincidence, the incomplete estate file of a Timothy Goodman from Gates County includes a note on one of the pages about “the inventory of the estate of Joel Goodman.” It almost looks like Timothy and Joel are the same person, but there is nothing else to indicate this. Still given that Timothy was given as a middle name to boys in more than one generation of Goodmans, it may be that there was a Joel Timothy Goodman.

So, to recap: there was both a Timothy Goodman and a Joel Goodman in Dobbs and/or Gates, NC, and apparently a Timothy Goodman and a Joel Goodman in Laurens, SC as well.
Throwing another Joel into the mix: there was a Joel Goodman, son of another John Goodman, in Stewart County where Timothy sold the above mentioned land! However, he was born in 1843, a decade after my Timothy sold land for the previous Joel.
Looking at all these Goodmans and their pernicious habit of re-using names, I think this is going to be a project on the level of the Dicken/Dickens family to sort out! I may have to circle back around to it at a later time.