The first sentence, "On one miserable morning, I loaded my kids into the car" is the better choice as it is more commonly used and conveys the intended meaning more accurately.
The difference between "loaded" and "moved" in this context is that "loaded" specifically refers to putting the kids into the car, whereas "moved" could refer to any type of movement, such as physically carrying them or moving them from one place to another. Therefore, "loaded" is the more precise verb to use in this sentence, as it conveys the idea that the speaker is getting the kids ready for a car trip.