You have correctly understood the major basis of Calhoun’s states’ rights doctrine. Calhoun does believe that the Constitution was a covenant between the various states and the federal government. However, this is not a clear and obvious misunderstanding of the Constitution. One can easily argue that it was a correct interpretation. The Constitution was not ratified by a vote of all the people. There was no requirement that 51% or 66% of the people of the entire country vote to ratify it. Instead, it was to take effect when conventions in nine of the thirteen states voted to ratify it. In this way, there is a strong argument to be made that it was a contract between the states and the federal government.
Calhoun then went on to argue that his interpretation of the Constitution meant that states had the right to nullify laws made by the federal government. They could do so if they found those laws to be unconstitutional. They also had the right to secede from the Union since they had entered it voluntarily and there was nothing in the Constitution to say they could not leave it.
Jackson did not have a fully-fledged interpretation of the Constitution like Calhoun did. He simply thought that the federal government needed to be sovereign. He felt that the country would break apart if this were not the case and he did not want that to happen.
You have, again, understood Calhoun’s position well, but it is important to note that his was not an absurd argument based on a clear misunderstanding of the Constitution.