When the diverter is in shower mode there is a steady trickle of water coming out the underside of the diverter face. It doesn't leak in tub mode. Could water be leaking into the wall also - is the only way to tell to take off and replace the whole diverter?

It sounds like you have a leak at the end of the shower arm which goes inside the wall, get a flashlight remove the Shower Arm Escutcheon Plate see the image below, turn on the shower, check inside the wall particularly the end of the shower arm and see if it leaks, if it does remove the arm apply some Teflon tap install and see if it is fixed, good luck.

John