There doesn't seem to be any significant wear on the gears / dog clutches within the diff. Everything stops turning with the clutch pedal pressed, but I *think* what's happening (given the way it works OK with the engine stopped) is that pushing the clutch isn't fully releasing tension on the transmission (and therefore the input shaft to the diff). With the engine running, if I really haul on the forward/reverse lever while "tickling" the clutch pedal, it'll usually work - but I suspect that I'll eventually break something doing that...
Question is, are there any "usual suspects" to look at? I can adjust the rod which connects the clutch pedal to the arm containing the dual-cone drive clutch (which serves as the transmission's gearbox) so that it travels further and therefore slackens the belts more - but I'm reluctant to mess with it if the problem's likely elsewhere, as I assume that messing with that will also result in belts potentially slipping during normal drive.
Another possibility: The gearbox side of things operates fine, which means that the conical clutch is able to slide up and down on its shaft as it's supposed to, but maybe this isn't sliding as well as it should (and so although tension is removed on the belt which links the engine to the clutch, it's not being removed adequately on the belt which links the clutch to the diff)?
(the mower's an old MTD/Craftsman with 38" deck and 11HP B+S vertical- shaft engine, not that it probably matters as I bet the drive arrangement's typical for all sorts of lawn tractors, even if the parts are slightly different)