During the lead up to and actual secession, what did all the members of congress from the states which had seceded do? Did they head off to the Confederate congress? Did they continue representing their states and acknowledge some as rebels?
As each respective state voted for secession, its members delivered a letter to Congress announcing that they had vacated their seats. So, for example, following the secession ordinance of Mississippi on January 9th, the following was entered into the Congressional Record for the House of Representatives:
The Speaker, by unanimous consent, laid before the House a communication from O. R. Singleton, William Barksdale, John J. McRae, Reuben Davis, and L. Q. C. Lamar, members of this House from the State of Mississippi, stating that "Having received official information that the State of Mississippi has passed an ordinance through a convention representing the sovereignty of the State, bearing date the 9th instant, by which ordinance she has withdrawn from the federal government all the powers delegated to it at the time she became a member of the federal Union, it becomes our duty to lay this fact before you, and withdraw ourselves from the further deliberations of this body."
Once these Congressmen had resigned, there was a debate about what to do about their vacant seats. Some argued that the states did not have a right to secede, and therefore their departure represented a kind of resignation. Others argued that secession was legitimate, and the seats held by the secessionists no longer existed. Finally, the Senate passed a resolution in March 1861 declaring the seats of the departed colleagues "vacant" while striking their names from the Senate roll. Ten more Southern members were expelled in July 1861.
A handful of other Congressmen were also expelled for disloyalty during the war, including two members of the House of Representatives of Missouri and one member from Kentucky (both border states which did not secede but were home to a large number of Confederate sympathizers) - in all, seventeen Congressmen were expelled in this fashion.
Some of these Congressmen were later elected as representatives of the Confederate Congress, which first assembled in February of 1862. For example, Robert M.T. Hunter, who had been a Virginia Senator in US Congress from 1847 to 1861 (and had ambitions to be President, only to be defeated by Stephen Douglas in the Democratic nominating convention), had urged his state to pass a secession ordinance even while he was a sitting member of Congress. He was consequently expelled from the Senate, but was later appointed Confederate Secretary of State and then successfully ran for Confederate Senate in 1861 and remained in the position until the end of the war.
Hunter's trajectory was not unique; most of the members of the 1st Confederate Congress were veterans of the United States Congress.