Yes you can drive into the south of Kafue (Dumdumwenze Gate) from Livingstone. You take the Lusaka road and turn off at Kalomo. But you need a good 4 x 4. There are two routes once past the gate, the eastern route is often impassable in the wet season, the other can have very deep sand. The gate will be able to advise. It will take you at least two full driving days to go the Lusake/Monze road which intersects the park (we took 3). You can sleep at places like Nanzhila and Kaingu en route. Biggest headache is fuel - supply at Kalomo is unreliable after that only option is buying from Zesco in Itetsi Tetsi. Expensive and ours was dirty. We had a 120L tank plus 40L in jerry cans which is just enough to get you from Livingstone to the next fuel on Lusaka/Monze road at Mumbwa.