But is there a way to diminish both the power of the state and the wealthy at the same time? It seems that at least, the best societies are those in which the state and private interests comete for power, but is there anything possible beyond liberal democracy that doesn't grant a monopoly of power to the wealthy or the political class?
Currently, I doubt it. The fact that we have limited resources - be they food, raw materials, land, material goods, or whatever - makes competition for them a necessity, and as such power is sought to help control those resources.
In the future as our technology advances, it may become possible. If we had the technology to automate the economy and enough raw resources to provide a comfortable lifestyle to everyone, you could possibly achieve a
post-scarcity society. In that case people's physical needs and most of their wants can be met easily, which may very well make the need for most competition obsolete. Wealth would be irrelevant because everyone would have a wealthy lifestyle, and government would largely just be focused on some basic administration and law enforcement. The latter focus would be decreased, as crimes involving theft wouldn't be as commonplace since theoretically nobody would have any need to steal.
Of course that's all theoretical - it may not be possible to reach that state.