I'm also a descendant of Washington Lindsay Coleman of Meade County Ky through Josiah Coleman. I have two DNA tests, one on FTDNA (Y-Chromosome) and one on Ancestry. The Ancestry DNA test shows 60+ DNA cousins that descend from every one of Josiah's known children - so a pretty clear indication that Wash was a grandson of Josiah.
Researching DNA cousins that are also related to known descendants of Josiah, I've come up with a few theories:
1) James Jeremiah Coleman/Elizabeth Leake. My great aunt, who is a great-granddaughter of Wash Coleman, had a DNA test done through Ancestry, and she shows 82 DNA cousins descended through this couple. Using triangulation, it appears that all of these cousins have Coleman ancestors. James J. Coleman's parents were Samuel Coleman and Anne Mourning Christian . My great-aunt is related by DNA to over 100 cousins descended from this couple, all of whom appear to be Coleman descendants.
2) Henry Coleman/Mary Hutcherson. My great-aunt's test shows 24 cousins descended from this couple
3) John Coleman (1757-1837) and Mary "Mollie" Coleman. There are about 20 DNA cousins descended from this couple. They emigrated from Virginia to Kentucky about 1809, around the same time as Josiah. Also Mary Coleman's mother's maiden name was Lindsay, a possible source of Wash's middle name.
Unfortunately there is no documentation that I've found that can help make a determination if any of these couples are Josiah's parents. The likeliest, based on DNA evidence, is the first couple. However James Jeremiah's will doesn't list Josiah, but lists many other children. That could be because Josiah would have been a minor at the time he died (around 13), but you would have thought the will would have left a provision for minor children in it, which it doesn't.