+1 about using D-A shifter or derailer with non-D-A derailer or shifter, respectively. Doesn't work.

If you remove the shifter from the frame, there is typically a tiny hole drilled in the back side that has a peg partly filling it. Add several drops of light lubricant like Tri-Flow to be sure the indexing isn't fouled with corrosion. The click action should be very crisp, and free-play is normal.
Don't use WD40, it's too thinned with kerosene to offer good lubrication.

The cable wire needs to positively be seated in the groove in the derailer's actuator arm (where the anchor bolt pinches the wire). Accurate indexing depends on this.

Also tug on the exposed cable wire along the down-tube, to check for smooth take-up and return of the derailer movement. Should be really smooth if the rear cabling is in good order and the guide is lubricated.

I kind of doubt that any parts are bad here, cept for maybe the cable.