I'm agency for an Ohio prison, but even for nurses moving from agency to state, it's a long process! One of my coworkers just got notified of her acceptance to become a state employee 3 weeks ago, but she will not begin orientation until January. Luckily, she can stay agency in the meantime.
Ohio seems to move slowly, imo - just the background check for me to work at this prison took 5 weeks.
If you don't get the state position, ATC Healthcare is the agency that holds the contracts for prison nurses in the state.