I searched for a while, but could not find any information on a marriage between Jane Latham and Andrew Goodson in North Carolina or Tennessee, however, there was a Jane Latham, born abt 1800 in Tennessee, living with Charlotte Goodson in 1860 and 1870 in White County, Tennessee according to US census records. Also, on the 1850 census in White, Tennessee, "Jane Goodson" also born abt 1800 in Tennessee, who was living with Andrew and Charlotte Goodson before Andrew's death in 1856.
It would seem likely that this Jane Goodson is the same Jane Latham in the two later records (unless two women of the same age with the same first name decided to trade off living with the same family). Perhaps the census taker just mistakenly grouped Jane Latham in with the family surname on the 1850 census? Or she was related to the family and then she married and changed her name, or someone died and she decided to go back to her maiden name? Or perhaps Jane and Charlotte were sister wives at the same time until Andrew died? lol. It's all a bit confusing.
I also found a death certificate in White, Tennessee for what I believe to be Andrew and Charlotte's daughter, Rebecca (Goodson) Swindell (22 Mar 1848-12 Apr 1930), which lists her parents names as "Anda Goodson and Charlottie Jenney." The death certificate was filled out by her son though, so that information is dependent on how well-informed he was. I searched for death certificates for all the other children to no avail. All of their Tennessee marriage records do not list the parents names.
1850 Census
Andrew Goodson, Age 51
Charlotte Goodson 21
John L Goodson 15
Jane Goodson 50
Rebecca Goodson 2
Elizabeth Goodson 1
Elizabeth Cobb 60 (also tried to find out how/if she is related)
1860
Charlotta Goodson 27
Rebecca Goodson 12
Mary E Goodson 10
Mitchell Goodson 9
Louisa Goodson 6
Jane Latham 60
1870
Charllotta Goodson 35
Rebecca Goodson 22
Elizabeth Goodson 20
Michael Goodson 18
Louisa Goodson 16
Jane Lathum 60