Turing Reduction vs Karp Reduction
The huge difference is that a Turing reduction allows you to call the black-box more than once. Suppose you have the $\mathtt{GraphColor?(G, k)}$ decision problem which tells you if the graph $G$ can be colored with $k$ colors. A Karp reduction only allows you to call this problem exactly once. But a Turing reduction allows you to call $\mathtt{GraphColor?(G, k)}$ a polynomial number of times. So if you want to figure out the chromatic number of the graph, call the function with $k=1, k=2, \ldots$ and so on until you get something that works. To make it more efficient, use binary search instead.