Not all of them, no. The 37th Congress contained delegates from several of the seceded states, including two from Louisiana, three from Virginia, and five from Tennessee. Most of the seats were vacant, however, and among the Confederate states only Virginia and Tennessee (until 1862) had any senators in office.

The members of the 36th and 37th Congresses representing states that seceded withdrew from Congress at different times. As an example, Mississippi seceded on January 9, 1861, but most of its Congressmen served until January 12, and Jefferson Davis did not withdraw until January 21. Some stayed through spring.

Senators of the 37th Congress who supported secession were expelled from Congress. For example, Senator Wigfall of Texas withdrew on March 23, but was (among others) formally expelled on July 11, 1861.

Edit: the linked page has a handy search function, and combined with the lists of Congressmen found on Wikipedia will tell you what happened with any given Congressman.

That's a large group of people, eight-eight in total. Some of them participated in last-minute negotiations, with varying degrees of real commitment to finding a solution. I discuss those in a previous post.. Without going through all the votes that took place in the House and Senate between November 6, 1860, and May 20, 1861 (the secession of North Carolina) and checking the roll calls, I can't say for sure that everyone who ought to have abstained (given secession) did. But the Congress got pretty busy in that interval and most of its business was about saving the Union short of bloodshed. It's all knowable information, but not a survey I've done or can easily do quickly.

But we can look at some of the men involved. Senators are much more prominent in the literature than representatives, so I'll go with them. There's also only twenty-two, which is a bit easier to manage. I'm going to do them in order of secession ordinance. For context, Lincoln is elected on November 6. The 37th Congress first meets, in special session, on March 4 (also the day Lincoln takes office) but the 36th Congress is on hand from December 3, 1860 until March 3, 1861.

South Carolina

Date of Secession: December 20, 1860

Senators were James Chesnut (Mary's husband), and James Henry Hammond. Chesnut withdrew from the Senate on November 10, 1860. Withdrawal did not mean resignation, necessarily. The Senate expelled him on July 11, 1861 because of the treason. Hammond withdrew a day later but doesn't seem to have been expelled. I'm not sure what happened there, but to judge from his biographical sketch (linked in his name) he must not have taken up political or military office with the Confederacy. The sketch says he withdrew, but he may have actually resigned. The difference is fairly academic. Neither South Carolinian went to Washington to claim their seats in even a pro forma manner.

Mississippi

Date of Secession: January 9, 1861

Senators were Jefferson Davis and Albert Gallatin Brown. Davis withdrew on January 21, 1861. I'm unsure what the cause of the delay there was, but I know that when he left he referred to instructions from the state government. He might have waited on those. The Senate voted his seat empty on March 4, 1861. By then he was president of the Confederacy. Brown withdrew on January 12, which raises further questions about just what kept Davis. (I honestly don't know, beyond that Davis is generally regarded as a hard guy to figure.) His seat was declared vacant on March 14.

Florida

Date of Secession: January 10, 1861

Senators were David Yulee and Stephen Mallory. Yulee withdrew on January 21, the same day as Mallory. The senate voted Mallory's seat vacant on March 4, but not Yulee's.

Alabama

Date of Secession: January 11, 1861 (rough three days here)

Senators were C.C. Clay, Jr. and Benjamin Fitzpatrick. Clay withdrew on January 21, seat vacated by the Senate on March 14. Fitzpatrick went on February 4. His seat doesn't appear to have been vacated and neither man was expelled. But they're not around casting votes after this, all the same. That goes for the other withdrawals too.

Georgia

Date of Secession: January 19, 1861

Senators were Alfred Iverson and Robert Toombs. Iverson left January 28, nothing further. Toombs withdrew on February 4 and had his seat vacated on March 4.

Louisiana

Date of Secession: January 26, 1861

Senators were Judah Benjamin and John Slidell. Benjamin left on February 4, seat vacated March 14. Slidell the same day as Benjamin. Appears his seat was not vacated.

Texas

Date of Secession: February 1, 1861, referendum to ratify on the 23rd. (Sam Houston dragged this out as long as he could.)

Senators were John Hemphill and Louis Wigfall Hemphill appears to have technically remained in office until expelled on July 11, but he represented Texas in the Confederate Congress so it's not like he was hanging around DC. Wigfall withdrew on March 23 and was expelled on July 11.

Sumter surrenders on April 12. Lincoln calls for men to suppress the rebellion on April 15.

Virginia

Date of Secession: April 17, 1861, referendum to ratify on May 23

Senators were RMT Hunter and James Mason. Hunter withdrew on March 28 and was expelled on July 11. Mason withdrew and was expelled on the same dates as Hunter.

Arkansas

Date of Secession: May 6, 1861

Senators were Robert Johnson and William Sebastian. Johnson's term ended on March 3. No action seems to have been taken against him. Sebastan was expelled on July 11.

Tennessee

Date of Secession: May 7, 1861, referendum to ratify on June 8

Senators were Andrew Johnson and Alfred Osborn Pope Nicholson. Johnson chose to stay in Washington and the Senate until March 4, 1862, at which point he resigned to be Tennessee's military governor. Nicholson withdrew from the Senate on March 3, 1861 and was expelled on July 11.

North Carolina

Date of Secession: May 20, 1861

Senators were Thomas Bragg and Thomas Clingman. Bragg withdrew on March 8 and was expelled on July 11. Clingman held on until March 11, then withdrew and was expelled on July 11.

That's all of them. One senator stayed in Washington, his state be damned. Omitting Robert Johnson where it looks like the state didn't bother electing a replacement (sometimes states could be quite tardy in doing that), we have seven Senators who quit in advance of their states (both SC, VA, 1 TN, 2 NC) and eleven who quit after secession was done (both of MS, FL, AL, GA, TX, 1 AR). Of all those who left, nine (1 SC, both TX, VA, 1 AR, 1 TN, both NC) were later expelled.

But since I'm here, there are a few more expulsions. John C. Breckinridge (KY) was expelled on December 4, 1861. Trusten Polk and Waldo Johnson (both MO) were expelled January 10, 1862. Jesse Bright (IN) is the last out (February 5, 1862) Bright took his expulsion so well that he tried to win the election to replace himself. He lost.

In any event, it's clearly not the case that people acted with perfect immediacy. Some of them jumped the gun but more of them acted after their states left. In most cases the delays look relatively reasonable given the speed of communication and logistics of picking up and going. Probably a few hung around to give farewell speeches or otherwise settle affairs, but even where there's a discrepancy between the senators of the same state departing it's not a huge one. Nobody stuck it out for another few months before deciding to go home, for example.

EDIT: Corrected a postal abbreviation error.