The spelling of the name from Gaulding to Gaulden/in occurred in the generation of the children of John Matthew Gaulding. Son John, Jr. and son William have been set to merge into their EN or IN derivations. Further research is needed to determine the derivations or not, used by the rest of the children. Some are known to have kept the Gaulding spelling.
The Gaulden, Gauldin, Gaulding, Family History -- A Seven Hundred Year Study, Volume One, p.30 Dr. Charles H. Gaulden, Self Published, April, 1999.
It may be possible to confirm family relationships with Josiah by comparing test results with other carriers of his ancestors' Y-chromosome or mitochondrial DNA.
Y-chromosome DNA test-takers in his direct paternal line on WikiTree: