Are they really sovereign though?

Consider Brexit. When the UK wanted to leave, the EU said "we are sad to see you go, please reconsider, but if you really want out, we can't stop you". There was some negotiation over the terms of leaving, but there was never any doubt that the UK had the right to do so. And it was always made clear that if agreement could not be reached on the terms of leaving, the UK would leave anyway, under "default" or "no deal" terms that would be rather unsatisfactory to both sides. (Prior to 2009’s Treaty of Lisbon, the EU treaties didn’t officially permit withdrawal, but almost certainly if some state had sought to withdraw pre-2009, they would have amended the treaties to allow it.)

By contrast, the mainstream position is that US states are not allowed to leave – at least not without the consent of the federal level (Congress), and many say it would even require a constitutional amendment (which would mean the consent of most of the other states would be required as well). So there is a very clear sense in which EU member states are sovereign (they are free to leave) but the US states are not (they have to ask permission to leave, which may well be denied.)

The Civil War really settled this when a group of states tried to leave and their attempt was forcibly suppressed. Now, what complicates the matter is their reason for leaving was to protect the utterly odious and reprehensible institution of race-based slavery. But, there is no necessary connection between the issues of slavery and secession, it is just a historical accident the two got linked.

Some people who opposed secession, and supported the Union side in the War, did so primarily because of their opposition to slavery. And if you imagine some alternative history in which a group of states (whether the same group or a different one) seceded over some other more defensible issue, many of those people might have not opposed secession, or at least not so strongly. But, on the other hand, other people who supported the Union side, their primary concern was anti-secessionism rather than slavery, and they would have opposed secession over some other more defensible issue just as strongly. A secession over some other issue might still have led to a civil war, which could easily have had the same anti-secessionist outcome.

Finally, whatever "sovereignty" the US states have is really at the mercy of the Supreme Court. The Supreme Court is free to interpret the notion of "sovereignty" as broadly or as narrowly as it wishes, and a future Supreme Court could even turn it into a dead letter, a purely theoretical notion – and since they appoint the Supreme Court, the President and Senate have the ability in the long-run to influence the Supreme Court's positions. In recent times, the Supreme Court has been majority conservative-leaning, and conservatives are probably somewhat more sympathetic to the notion of "state sovereignty" than liberals/progressives are – but, I think the current conservative majority is mostly just a historical accident, the conservative side got lucky, it has had a liberal/progressive majority in the past and could well again at some point in the future.