Incorrect.

Socialism has the state, because socialism isn't global, and still has private property in the process of being sublimated from the private sector into the public. You cannot simply stroke the pen and legalize full public ownership, the conditions for social ownership and planning need to be developed and built gradually. As there is still private property, there is still the bourgeoisie and proletariat, and thus the proletariat contains ownership of the state to oppress the bourgeoisie.

Communism exists when all property has been sublimated. There's no need for the elements of society that existed under socialism to keep the bourgeoisie in check, so these wither away, but the development of large scale industry that necessitates administration, management, and planning continues to exist. There's no state, but there remains "government."

Marxism is anti-utopian. Communism is not a utopia, it isn't a formula to create outright. Communism is compelled economically through the development of socialism. As management and administration remains, there is still equal, collective ownership. Complex methods of democracy are built up during socialism, it would be economically impossible to have direct democracy for every single decision to be made in a global economy, hence the need for social planning, administrations, and managers.

You are making a dramatic error by conflating anarchist analysis of hierarchy with Marxist analysis of class, and thus you confuse what Marxists want with what anarchists want. Unless your point is that Marxists aren't communists, of course, but that's just deeply silly.

I really suggest you actually read Marx if you want to speak as though you know what Marx actually advocated for, rather than just assuming Marx was an anarchist with different methodology.