At any point did states or even counties or towns purchase slaves for their own use? For instance, these days you have public works departments that belong to towns or counties that maintain roads, parks, and various other things that need upkeep. Did towns and cities as entities purchase slaves for this use?
I can't say that no one ever did it, but not to the best of my knowledge. States and municipalities do use enslaved labor, often for public works projects, but those slaves are "hired". For hired, read: their enslavers sell their labor and pocket the fee. When slaves come into government possession, because someone owes money, dies intestate and without clear heirs, doesn't pay their taxes, or they're illegally imported, those enslaved people are state property in the technical sense but they're not generally put to work on state-owned plantations or such. Instead the goal is to auction them off and convert their labor directly into cash for the public coffers. Let me give you an example of how this shakes out. The specific cause here is a little exotic -illegal slave importation- but the general frame applies to other situations where property reverts to or is seized by the state.
The full story is here but the long and short of it is that David Brydie Mitchell, just-former Governor of Georgia, took a job as Indian Agent to the Creek Nation. It paid better and might last longer than another term in the Georgia Executive. Being out on the frontier of white settlement, he was far from prying eyes and also relatively near Spanish Florida. He and some associates promptly got together to smuggle in enslaved people via Amelia Island and St. Mary's River system. They got caught red-handed with around 110 people.
Most of those enslaved people (88 total, the others probably ended up in Alabama) were seized by a local military command under the authority of the surveyor and inspector of the Port of Darien (same person) and his brother the collector of customs. Because of how the slave importation prohibition law is written, the federal officials here can seize the slaves but have to deliver them to state authorities for final disposition. The fear was that if they were left in federal hands, they might be freed. It's one thing to bring slaves into the country illegally, but quite another to free them once they got here.
Those enslaved people belong, legally, to Georgia. Georgia doesn't keep them up, though. After some delay to comply with a mandatory period of public notice, the state sold the slaves for $34,736.18. The expense of housing them with people about Milledgeville in the interim and other liabilities brought that down to $27,571.82 in profit. During the public notice period, people could have tried to prove legal ownership and just take them back, probably paying some of the expenses of their room & board as part of the deal. That was also the case when enslavers used public jails or public officials to manage their slaves on a temporary basis.