"On 25 Apr. 1794, it was reported to the Town Council of Cranston that Olney Waterman is absent and left his wife and children so they are likely to be a public charge. Nathaniel Carpenter, "the present overseer of the . . ." (The Waterman Family, Vol 3, p 117; by Edgar Francis Waterman, 1954)
Olney's sons John P. at age 9 and William at age 3 were apprenticed in March 1794; John was apprenticed to William Sprague of Cranston, who died in 1795, so after that he was apprenticed to William Sprague's son of the same name.
I haven't found Olney Waterman in the 1790 census in RI, but there is one in Chester, Windsor, VT.