should I replace the drain and P trap

If that is what is leaking, yes. But you need to make sure what is leaking first. If it is a shower only, the usual cause is the rubber compression washer/nut just needs to be tightened or replaced. There is usually a nut that screws in from the top and compresses the rubber o-ring tightly around the pipe. Over time that may loosen. It does not look like a nut, but may only have a couple of bumps on it. Tapping those bumps in a clockwise direction should tighten it.