The transconductance and so essentially the speed of the NPN series pass transistor in the LM338 is proportional to its output current, hence the asymmetry: it can charge the output node up fast, but the node is only discharged by the 5 mA min load + quiescent current and output voltage setting divider current... so pulldown is slow.
That doesn't really explain why it's underdamped. But I think that is also related to the asymmetry: the output stage is very slow when the current is low, so the output pole is closer to the internal dominant pole, reducing the phase margin when voltage is dropping, leading to undershoot that is then corrected with another overshoot etc.