It's not acceptable for the State DOT to simply divert water onto yours or anyone's property, and expect you to pay for water management. Do they have a water easement on your property. If not, then I would contact legal help to petition the State to assist you financially in this water management project. I am only thinking of the excessive and disproportionate cost to you. Because this could get very expensive, to manage water runoff in these high volumes and avoid serious surface erosion issues.
Perhaps diverting the runoff to a nearby surface tank/pond to hold and retain the water during summer months, while it recharges during late winter melt and spring rains. These ponds can often be built with a bulldozer in a week long effort, and hold 300,000 gals or more. You will need to get a reasonable estimate of runoff in gallons, if you decide to go the pond option, because that will control the size of your retention pond. Such a pond can be lined with impermeable clay for year long retention, or permeable compacted sand for allowing ground water recharging.
Honestly, any flow channels you build should be lined with concrete to limit surface erosion and insure long term operations. And that's where the costs go up significantly.