IF having exchange – according to valuations – was a market, then every economy would be a market economy. What does exist in every economy is allocation. This is determination of what is produced, by what means, in what quantities, and to whom.
Unless one has an unknown form of economic dictatorship in which some person or group simply decides all that, with no valuations, etc. – one has exchange of the sort that is troubling you. But that doesn’t mean one has a market.
Markets for allocation means allocation via competitive choices by separate market actors each out for him or or herself and each paying attention only to whether to enter, or not enter, a specific transaction, and so on.
Participatory planning has prices (relative valuations to use in making informed decisions) and has income (a share of the social product for all actors), but it arrives at these in a very different way than either markets or central planning.
Money is just a convenience – prices – the real issue – are a summary representation of how society feels about an item and all that goes into its creation and use. Exchange is unavoidable – and certainly not a negative – unless everyone simply forages for self and self only – rejecting exchange says we should all operate in isolation.
The trick is to have valuations (necessary for making sensible choices) and to have exchange (necessary for any kind of social connection as well as sufficient output to survive and thrive) that in fact reflect true social, personal, and ecological costs and benefits, and to use them in ways that foster values one believes in – for example self management or classlessness…
Beyond that, I would recommend taking a look at one of the full presentations of parecon. Then, if you have further questions, I would be happy to try to help.