The 1877 House Journal appears to have a misprint for Harrington in Wells County. The 11/29/1876 Indiana State Sentinel (of Indianapolis) gives Harrington 38 votes in that county.
The stated total for Harrington is 12,710 but the county figures add to 12,707.
James Williams would die in office on November 20, 1880. He was succeeded by Lieutenant Governor Isaac Gray.
Source for Popular Vote data: Indiana General Assembly. Journal of the House of Representatives of the State of Indiana, During the Fiftieth Session of the General Assembly, Indiana General Assembly (https://babel.hathitrust.org/cgi/pt?id=inu.30000061413252&view=1up&seq=59) (accessed 18 Oct 2023)