In the Freeholder Records available at the PRONI website, Lawrence Shoulder is listed as early as 1818 in Carron, on the land owned by Mr. Brownlow. The name is alternatively spelled as SHOULDERS.
This Lawrence must be the father to your Lawrence and grandfather to James if he was a tenant in 1818 (ie, of age) and the one who passed away in 1870, aged 88.
Lawrence Shoulder is in England by the time of the 1861 census, aged 25, in Willington, living in the home of Alice Gregory, noted as a visitor, married, born Ireland. He is married with Ann and a child Bridget in the 1871 census.
By 1881, the other sons have arrived -- surmise that Lawrence came first about 1861, leaving Anne and James and Owen at home in Ireland, who then arrived altogether about 1869, since the next child, Bridget, is born about 1869.