Karl Marx did not describe his ideal for after capitalism in much detail. This was largely because he didn't believe it was possible to know the result of instituting communism (not just in terms of political organization, at least if you interpret Engels as not too far from Marx in terms of standpoint, then most philosophical issues couldn't really be settled while in the throes of a society with class warfare, see "On Morality" in the Marx-Engels Reader for instance). One key feature though was that the state was supposed to play a transitional role in helping organize a transition to a stateless, locally run, society built around (most likely) worker's organizations. This is evident in the revolutionary organization of the USSR, the term "Soviet" refers to worker's councils that were democratically run, but these were largely sidelined in the early years of Lenin's time as leader. A simple source for the timeline of events, if not detailed analysis, is the piece "The Bolsheviks and Worker's Control" by Chris Pallis.

Certainly a long-lasting centralized (economically or politically) undemocratic state was not something Karl Marx prescribed.

There are aspects in which the USSR did show an interesting fidelity to certain leftist ideas. For instance, their approach to "the National question" does seem to be guided at times by earnest engagement with Marxist ideas. This explanation by u/theshadowdawn seems right to me. The same could be argued for the Leninist New Economic Policy and the Stalinist collectivization policy. You'd be very hard pressed though to say that, e.g. Stalin's writings on Marxism in linguistics, were driven more by deeply held Marxist beliefs than by pragmatic opportunities to attack perceived intellectual opponents, in that case Nikolai Marr's disciples.

While for the majority of communist countries, there's been immense interest in how liberalizing reforms made these countries more integrated with the rest of the world, there are cases where reforms were towards an economic model that shared more traits with traditional socialist ideals, for instance, the Yugoslav policy of worker's self-management.