In 1746 he and his elder brother Thomas entered the dissenting academy at Kendal under Caleb Rotheram, as exhibitioners of the London Presbyterian Board.
In 1754 Dawson succeeded Gaskell as presbyterian minister at Leek, Staffordshire, but soon moved to Congleton, Cheshire, probably to assist in the school of Edward Harwood.
In 1763, being now LL.D., he accompanied a young Yorkshire baronet, Sir James Ibbetson of Leeds, to Warrington Academy as his private tutor, and joined the literary coterie of which John Aiken was the head.
Alexander Gordon writing in the Dictionary of National Biography considers that Dawson's conformity was a protest against the Arianism in fashion with the liberal Presbyterians of his time; and notes that Dawson's argument is stronger against Arianism than Socinianism.
In 1764 he followed Edmund Law in reducing the intermediate state to the sleep of the soul, and in 1783 he wrote strongly in refutation of the moral objections to the doctrine of necessity, against the language of the Articles.