This is actually a very common question on AskHistorians, and we have five answers in our FAQ here. This is the one I typically refer people do, courtesty of /u/Georgy_K_Zhukov, being thorough and comprehensive.

The essential answer is that you're a little bit right, and a little bit wrong, too. You are quite right in saying that the Northern states did not initially conceive of the war as a war to end slavery by force; the shift towards an anti-slavery agenda, though, comes quite quickly. Within a year or two of the war beginning Lincoln's administration already has its mind set on how to bring slavery to an end, though it does not yet adopt a policy of immediate and total abolition; in late 1862, Lincoln essentially issues a warning statement to the Southern states that he is preparing to issue the Emancipation Proclamation and that if they have any hope of preserving slavery, they would need to rejoin the Union immediately.

From the first day of 1863, the Union effectively made slavery's end a driving goal of the war through the Emancipation Proclamation; though the Proclamation did not abolish slavery, it provided for the freedom of the vast majority of slaves in the United States as a war measure, which in essence declared the Union's intention to decimate the institution of slavery. For much of the war, Lincoln and his administration were actively trying to conceive a solution to the problem of slavery that would secure the institution's end - his own initial preference was a policy of gradual compensation emancipation, whereby the government would purchase the freedom of all slaves. This was the policy that the British pursued in their colonial territories but in the United States, it would likely have proven financially impossible to reach a deal with the Southern elite, most of who had no interest in accepting as a matter of principle anyway (an attempt to secure compensated emancipation in Delaware had ended in utter failure).

So even if the federal government did not enter into the war with ambitions to pursue an abolitionist agenda, the experience of the Civil War quite quickly shifts its priorities towards the end of slavery. From at least 1863 onwards we can say that the Civil War was effectively being prosecuted in the interests of fatally damaging the institution of slavery as well as reunifying the nation, a reality that was not lost on the Southern elite, who simply saw the shift in priorities as confirmation of their underlying fears that the Republican Party was hell-bent on abolition.

but the Union didn't go to war with them over slavery, it was over the secession itself (aka, the federal government asserting its dominance over the states. In this case, their right to self-determination).

There are two fundamental problems with this argument. The first is that there is and was no legal mechanism by which the Southern states were entitled to secede from the Union. The Constitution does not provide for the sovereignty of any individual State, and explicitly afforded the federal government the power of determining the use of its territory, whilst at the same time requiring the United States to provide for the protection of its constituent members. Each of the Southern legislatures was bound by Article 6 of the Constitution to uphold its provisions, and decades before secession the Supreme Court had affirmed the supremacy of federal law over state law when the balance between the two is not constitutionally defined (see McCulloch v. Maryland). Even before the modern federation was formed, the Northwest Ordinance of 1787 was explicit in declaring that "the said territory, and the States which may be formed therein, shall forever remain a part of this Confederacy of the United States of America". In other words, even under the weaker Articles of Confederation, the now-states of Ohio, Indiana, Illinois, Michigan, Wisconsin and Minnesota would not have a legal right to secede - and those Articles did state the each member "retains its sovereignty, freedom, and independence".

You can make a moral argument that states ought to have a right to secede from the union at will, but the North was certainly acting in a fashion perfectly compatible with the constitutional order of the United States, and the Civil War cannot be legitimately interpreted through a framework of states' rights versus federal authority in the way that this argument implies. There is simply no strong legal or constitutional basis for secession even before the Civil War. This was affirmed after the Civil War in the Texas v. White case, which found that when "Texas became one of the United States, she entered into an indissoluble relation", and that secession is only possible where it is by mutual consent - in other words, a state perhaps has a right to request to leave the Union but the granting of that request is conditional upon agreement with the other states. One state could not unilaterally secede - no such right existed within the USA's constitutional fabric.

The other problem with this interpretation is that it is one of the key ways in which Confederate apologists seek to take away from slavery as the driving theme of the Civil War. They try to distract from the issue by recasting it as a struggle for states' rights, arguing that even if slavery is abhorrent, it was just the unfortunate issue over which the wider struggle for states' rights was fought (assuming slavery's significance is not almost completely excised from the discussion, which is very common with Lost Cause apologism). Unfortunately, though the Southern states do take up the language of ideas about states' rights to make their argument, any principled commitment to that value was always secondary to the interests of slavery as an institution. In the run up to the Civil War the Southern elite was not advocating for greater State autonomy - they were on the contrary demanding that the federal government be more involved in the government and laws of the States, in the sense that they expected it to do whatever it could to enforce the provisions of laws like the Fugitive Slave Act of 1850, which along with the Dred Scott case served to give slavery an institutional and legal presence even in the states that wanted nothing to do with it.

In other words, the Confederate states were interested in self-determination and autonomy only if it benefited the institution of slavery. Where the right to self-determination in law and government undermined that institution, they were very much eager for the federal government's powers to be enhanced. The inadequate enforcement of the Fugitive Slave Act in Southern eyes, combined with the resentment of several Northern states and their determination to undermine it, was one of the great grievances of the Southern elite who pushed for secession. When secession ordinances talk about the failure of the federal government to protect their rights, they are really talking about the failure of the federal government to satisfactorily put their sectional interests ahead of and at the expense of those of other states.

The 13th Amendment was unable to be passed in the House of Reps in early 1865 (nearly the end of the war) and when it was passed later that year, it was only by a tiny margin, which was facilitated by bribes

This point is perhaps slightly disingenuous and needs clarifying. The House only passed the 13th amendment by a tiny margin, yes, but this is because constitutional amendments require a two-thirds majority rather than a regular majority of half plus one. The balance of power in the House of Representatives was firmly in favour of the anti-slavery faction by 1864, and the Republican Party itself would have a two-thirds majority - and thus the numbers to pass a constitutional amendment - from March 4th, 1865. The final vote was 119 - 52, which is a huge majority for legislation in general, but narrow for a constitutional amendment. It would not have passed without the aid of rebel Democrats who were persuaded to cross the floor certainly, but an amendment in the next Congress very likely would have when more Republican lawmakers assumed their seats.