Well in the case of Massachusetts, Liberal courts saw a meaning in the newly written Massachusetts constitution that was not intended when the document was written, so they "interpreted" it to mean slavery was abolished. (Massachusetts sure loves themselves some dictator judges telling them what the law "means" in contradiction to what it actually means.) Other states' legislatures voted for laws to gradually abolish slavery, though many of them claimed to have "abolished" slavery, but had not actually abolished it because slavery was still going on in those states even though they claimed to have abolished it.
But the 15 slave states (16 when West Virginia became a state) still had slavery, meaning the United States still had slavery, and would continue to do so indefinitely.
So, you admit that slavery could end by means other than a constitutional amendment. That’s progress, sir! Let’s take the next step. Is it so difficult to imagine that the gentle folk sipping on mint juleps while relaxing on the veranda weren’t so confident in the future of their peculiar institution as you are today?
“We are sent to protect, not so much property, as white supremacy, and the great political right of internal self-control-—but only against one specified and single danger alone, i.e. the danger of Abolition rule.”
-Jefferson Buford, Barbour County, Alabama, speaking to the Alabama Secession Convention, on March 4, 1861