I thought the proper tool was two concentric sockets, to allow you to adjust cone and locknut and snug them up against each other without trying to use a screwdriver to jam the cone in place while with your second and third hands you tighten the nut.
^ Correct.
PD-M323 and PD-M324 both have a similar arrangement.
There is no tab washer between the cone and locknut, so trying to hold the cone with a screwdriver is unlikely to be sufficiently effective; the cone is liable to slip before the parts are tight enough.
Adjusting these pedals is doubly a PITA because you can't use the correct tools on them unless you remove the cage first. Obviously you can lubricate the pedals without disturbing the adjustment; if the adjustment is OK and you don't have the right tools I would suggest that you do that.
BTW if the cone and locknut are not tightened sufficiently against one another, they will tend to move in use. The likelihood of this (or how tight the cone and locknut need to be to resist it) varies with how hard you push on the pedals when riding. If movement occurs (via precession), the usual thing is that the LH bearing adjusts itself tighter and the RH bearing adjusts itself looser. Both are to be avoided, because either will wreck the pedals.
NB; also, these pedals have RH threads for both L and R pedal cones and locknuts. Some later SPD pedal designs have a LH threaded cone and locknut on the RH pedal instead.
cheers