I think that in this case, to "Sherman" means to refuse to run for office.
Despite his age, Franklin would probably be the second choice. He was the second-most prominent American, and he needn't be that active, as the early presidents had much less power than they do now, and the president was more of a caretaker role, making sure that everything ran smoothly.
Other candidates include John Jay, John Hancock, John Adams, George Clinton, Patrick Henry, and Samuel Adams.
Two very intriguing candidates would be John Paul Jones and Thomas Paine.
From hindsight, we know that the first president would set precedents for the following presidents. Therefore, whoever held the job first would take the presidency - and the nation - in his own direction. While Jay, Hancock, and J. Adams would be more or less similar to Washington, Clinton, Henry, Jones, and Paine would take the nation in an entirely different direction.