The Corwin Amendment was the result of the efforts of the Committee of Thirty-Three in the U.S. House, which had tried to negotiate a compromise that would keep/bring back the seceding states into the Union. Like all the other peace negotiations during the Secession Crisis, they were primarily concerned with coming up with a resolution to the issue of slavery's expansion westward, as well as the "Fugitive Slave" laws. Other issues negotiated included the future of slavery in Washington, D.C., and the issue of "slave transit" in shared waterways of free and slave states, and transit of slaveholders with enslaved people into free states for business.

Eventually, what the Committee settled on was a plan that would enter New Mexico as one large slave state, encompassing the entire area below the original "Missouri Compromise" line, up to the border of California. The purpose was to resolve slavery's expansion westward—the South would get an additional slave state, and all the territory on the Southern border minus California, though not as many slave states as they wanted.

A second part of the proposal was to call for the peaceful acceptance of the constitutional Presidential election results as "the paramount duty of every good citizen of the United States."

The final part of the Corwin Plan was the part that became the Constitutional Amendment. That was the only part of the plan that got out of committee. The rest of it never got out of committee, never was voted on by the full House, and never made it to the Senate.

Lincoln, along with other Republicans, supported the final amendment because he believed it didn't change anything already in the Constitution, as he mentioned in his First Inaugural Address:

"I understand a proposed amendment to the Constitution—which amendment, however, I have not seen—has passed Congress, to the effect that the Federal Government shall never interfere with the domestic institutions of the States, including that of persons held to service. To avoid misconstruction of what I have said, I depart from my purpose not to speak of any particular amendments so far as to say that, holding such a provision to now be implied constitutional law, I have no objection to its being made express and irrevocable."

In the book Lincoln and the Decision for War: The Northern Response to Secession by Russell A. McClintock, the author states that when the Corwin Amendment was passed in the Committee of Thirty-Three before moving onto the full House, "two radical Southerners withdrew in protest of what they viewed as a fraudulent pretense of compromise".

The first of the two McClintock is probably talking about is Reuben Davis of Alabama, who "ceased to be a member of the committee, having been excused by the House" on December 29, 1860. He would leave Congress on January 12, 1861, the day after Alabama seceded.

The second of the two is probably Miles Taylor of Louisiana, of whom it was recorded on December 29, 1860:

"Mr. Taylor remarked that the decision thus reached made it clear to his mind that there would be no agreement by the committee upon propositions for the adoption of amendments to the existing Constitution which would be effectual for the settlement of the issue now pending between the two great sections of the country growing out of the slavery question, and that, in consequence, it was his purpose to take no further part in the deliberations of the committee; and it was his determination not to vote upon any other proposition pending before it. He further stated that he did not propose to resign his place on the committee, as it was his intention to remain in a position which would enable him to join in a minority report upon the subject submitted to the committee, if that should become necessary or appear advisable."

Taylor left Congress on February 5, 1861, ten days after Louisiana seceded.

The work of the Committee of Thirty-Three was tabled on January 14, 1861. At that point, while there were still other proposals being debated on the floors of the House and Senate, there was a proposal outside of Congress for what became known as the Washington Peace Conference of 1861. This was to be a convention of elder statesmen from the Border States, to work out a compromise that would then be submitted to Congress. This effort at compromise convened on February 4, and adjourned on February 27, having come to no acceptable resolution.

The Corwin Amendment was then proposed again in Congress. It passed in the U.S. House and Senate on March 2, 1861, in the last 48 hours of the Buchanan Administration. However, by that time, seven of the Confederate states had already seceded. Considering themselves a separate country, they never considered the amendment. Of the other four Confederate states, three of their legislatures were out of session at the time the Corwin Amendment passed. They didn't reconvene until after Fort Sumter:

Arkansas's General Assembly adjourned on January 21, 1861.

Tennessee had adjourned in late 1860, but they held a special session that convened on January 7, 1861, to consider whether or not to have a Secession Convention Election. They passed that resolution, dealt with some other business, and adjourned on February 4, 1861.

North Carolina's General Assembly adjourned on February 25, 1861.

Virginia was the only future Confederate state that was in session at the time. The Amendment was read in the General Assembly on March 18, and printed and distributed to the members of their state senate, but that's as far as it went:

"The SPEAKER also laid before the house a communication from the president of the United States, enclosing a resolution adopted by congress in relation to an amendment of the constitution; which were read, laid on the table and ordered to be printed."

The Virginia assembly adjourned on April 4, 1861, without taking any action.

Kentucky actually did ratify the amendment that same day, on April 4. And Maryland did, too, about a year later, as did a couple of Northern states. But after Fort Sumter, it became a backburner issue until the war itself was resolved. And then once the war was not over quickly, it lost support all together. Instead, by 1864, the loyal U.S. states began efforts to pass the 13th Amendment outlawing slavery. The Corwin Amendment was effectively dead.

Thus, no Confederate state ever really took up the issue in their state legislatures. Rector's statement you quoted from might be in reference to the Corwin Amendment (his speech was made the same day the amendment passed in Congress), but I think he's referencing the failed Crittenden Compromise as well. That compromise was also a series of proposed Constitution Amendments, and those were friendlier to the South, but they had failed back in December. Anything less than the Crittenden Compromise, secessionists were generally unwilling to accept—and for many in the Lower South, even the Crittenden Compromise wasn't enough.

The governor of Tennessee, Isham G. Harris, made a similar statement read at the opening of Tennessee's special session that convened on January 7, 1861. While it wasn't aimed directly at the Corwin Amendment, Isham proposed his own compromise that was similar to the Crittenden Compromise. He also made lengthy statements about how the North wasn't upholding their Constitutional obligations. After detailing his proposed compromise, Isham followed up with:

"With these amendments to the Constitution, I should feel that our rights were reasonably secure, not only in theory, but in fact, and should indulge the hope of living in the Union in peace. Without these, or some other amendments, which promise an equal amount and certainty of security, there is no hope of peace or security in the government."

Rector's and Harris's positions, as well as Taylor's position in the Committee, were essentially that the South needed much more severe guarantees from the North than what the watered-down Corwin Amendment proposed. And without such guarantees, slavery was too much under threat to stay in the Union.