I can't say I envy you in this weather but you need to get up real early at Copper Breaks. I can appreciate that many of the local parks are still closed from the spring floods and the ones that have reopened are overcrowded from pent up desire so you have to cast your net a bit farther from home to get a decent spot.
I did have a nice shady one at Copper Breaks but w/o electricity. I stayed one night in April and the next at Caprock Canyon on my way to Palo Duro which I did not make and plan a do-over on that trip this fall. At Caprock a tremendous nighttime thunderstorm filled the canyon below my campsite, overflowing the bridges which took till noon the next day to clear, thus cutting short my time to get to Palo Duro.
If you've never been to Fort Richardson SP they have several nice sites along Lost Creek and a fairly extensive trail network for bikes.
Enjoy your trip and wear lots of sunscreen.
Here's some Copper Breaks pics:
M C