We moved from MI to OK nearly 30 years ago. For summer camping, OK is too hot for my taste. Shoulder seasons are better. Mid-March through April can be pretty good, and fall (Sept 20 to mid November) is even better camping weather. Peak tornado time is May, but honestly a person can find himself in severe weather just about anywhere in the country, just about any time of year. Bad weather happens. Even when a big storm spawns a series of tornadoes, the tornadoes are only affecting a tiny percentage of the state's total land area; it's sort of like being struck by lightning to be hit by a tornado. And OK doesn't have a monopoly on them, either; we hear of tornadoes tearing up places in many other states.
As for state parks, Red Rock Canyon might be the most unusual due to the reddish canyon walls surrounding the CG. Robbers Cave, Osage, and Greenleaf are some other nice ones. Don't write off the COE CGs, though, because OK has some nice ones of those. I camped twice this year at Flat Rock Creek.