False.
No it's not.
These events transpired on 12/30/60:
"That night, after the long argument had ended, Black made up his mind to to resign from the cabinet unless Major Anderson was given proper support, and his threat (supported, it was said, by Stanton and Postmaster General Holt) seems to have been decisive. The President finally made up his mind. He would deal no more with the South Carolina commissioners and he would not order the soldiers to leave Fort Sumter. Whatever might come of it, the administration henceforward would resist secession."
-- "The Coming Fury" p. 165 by Bruce Catton.
Walt
"There is little that I can add to letters and telegrams previously dispatched. We never had a chance to make Lincoln an offer of any kind. You can't negotiate with a man who says you don't exist."
The central barrier to a peaceful solution in 1861 was Abraham Lincoln himself. That much is an indisputable historical fact.
I think Buchanan had a number of cabinet members resign near the end of his term. A letter in The Daily Mississippian published January 16th, 1861, says:
Secretary Thompson to-day resigned to the President his commission as Secretary of the Interior, on the ground that after the order to reinforce Major Anderson was countermanded on the 31st of December, there was a distinct understanding that no troops should be ordered South without the subject being considered and decided on by the Cabinet. At the Cabinet meeting on the 2nd, inst. the matter was again debated, but not determined. Notwithstanding these facts, the Secretary of War, without the knowledge of Secretary Thompson, sent 250 troops in the Star of the West to reinforce Major Anderson. Not learning this till this morning, he forthwith resigned.