One Williams family, apparently headed by Simon Williams, arrived as early as 1812 in Gibson County, buying land in Smith Township, Posey County in 1814. That's the northeast corner of the County. They migrated from Warren County, Kentucky, having come from NC and VA, probably with a stop in SC and TN.
Another Williams family appears to have come from Woodford County, Kentucky. In the east and south of Posey County it looks as though there may have been two different Williams families.
So there were likely three or more Williams families between 1817 and 1819, at least two of which migrated from KY.
I haven't come across a Jesse Williams that was living in NE posey County.