It is you that is incorrect. The Amendment read: "No amendment shall be made to the Constitution which will authorize or give to Congress the power to abolish or interfere, within any State, with the domestic institutions thereof, including that of persons held to labor or service by the laws of said State." It would not have guaranteed slavery in the territories, even those where it currently existed, nor would it guarantee slavery in any territories acquired in the future. It would limit slavery to the 13 states where it existed and no further. It was in keeping with the Republican platform calling for an end to the spread of slavery.
The secession occurred because of a variety of reasons, federal expansion due to reapportionment being a large reason. This added to the Northern nullification of the Constitution as being evidence that the Republicans would not observe equal protection for the Southern states. Remedy? SECESSION!.
By far the single most important reason for the Southern rebellion was defense of their institution of slavery. Take away all other reasons, but leave slavery, and the South still rebels. Take away slavery, but leave every other reason you care to name, and the South does not. It's as simple as that.
None of the formal declarations of secession mention that slavery caused their actions, despite what you may want to insinuate.
None of them mention any reason at all. But if you read the four declarations of the causes of secession then it's clear that the threat to slavery was above any and all other reasons given. They identified themselves by slavery, they depended on slavery, they identified their foes by their position on slavery.
For example, here's a statement from the second paragraph of A Declaration of the Immediate Causes which Induce and Justify the Secession of the State of Mississippi from the Federal Union:
"Our position is thoroughly identified with the institution of slavery - the greatest material interest of the world."
It was all about slavery, an unworthy cause. The unworthiness of that cause and the lack of sustained endurance it inspired in the Confederate public was just one of the fatal illnesses that doomed the rebellion to failure.
I looked up some of the proceedings in Congress where the amendment was passed and found that the following additional amendment was proposed in the Senate on March 2, 1861, but defeated 28 nays to 18 yeas [Source]:
Under this Constitution, as originally adopted and as it now exists, no State has power to withdraw from the jurisdiction of the United States; and this Constitution, and all laws passed in pursuance of its delegated powers, are the supreme late of the land, anything contained in any constitution, ordinance, or act of any State to the contrary notwithstanding.