Lamesa History 1922. Lamesa, the county seat [of Dawson County], has had a steady growth since the organization of the county, which was in 1905. It has four churches, two banks, five grocery stores, three dry goods stores, three drug stores, two hardware stores, one variety store and confectionery, two millinery stores, one electrical supply store, one harness and saddle shop, two abstract plants, four hotels, two restaurants, two haberdashers, three garages. three blacksmith shops, four gin plants, one ice, light and power company, one watchmaker, one butcher shop, three petroleum agencies, and preparations are being made to erect a seventy-five thousand dollar school building. - History of Texas, 1922, by W. Barrett Travis.