Marcia, Thanks for your interest in the Twitchells. Unfortunately you may have a different Moses Twitchell in mind than the several Moses Twitchells of Maine. I think, after some preliminary glimpses, that your Moses (father of Ada) was born in Sullivan County, New Hampshire. I have not found him in the Twitchell Genealogy by Ralph Emerson Twitchell, but circumstantially, I would guess he is either son of or brother to an Eli Twitchell of Unity, Sullivan, New Hampshire in the 1840 census (and that name is repeated in Ad's brother's middle name). Beyond that, I can only point you to rootsweb.com and recommend you search the family trees where you should find a tree containing Mary Mann as wife of Moses Twitchell.
Please let me know if you find more, as I am interested in finding out how this Moses connects up with the Twitchell genealogy.