after running the dishwasher or just running water into the sink, about 30 seconds later goes this nerve wrecking: POCK... POCK... POCK The dripping comes from the garbage disposal pipe and the water drops are falling into the P-Trap below. The figuration worked for over 10 years. What changed?
Your drain vent has become blocked.
More likely, based on your statement that the figuration worked for over 10 years, the vent has been absent, incorrectly installed, or blocked for over a decade, but the sink drain was somehow getting vented anyway.
If you recently changed any drain plumbing elsewhere in the house, you must have inadvertently eliminated the incorrect but accidentally working sink drain vent.
If you have made no plumbing changes anywhere, then the drain pipe must have originally been large enough to vent itself, but has been steadily, year by year, becoming narrower and narrower due to accumulation of grease and other obstructions.
In any case, your problem is a lack of proper drain ventilation.
Dripping water goes tick... tick...tick... not POCK! POCK! POCK! And how did you figure out where the water was dripping from and to, inside an opaque drain pipe? The loud noise you hear is not water dripping. It is air being sucked through the P-trap by water flowing through a choke point somewhere between the trap and the next downstream vented pipe.
The effect of a blocked drain vent goes beyond the nerve wracking noise. When the water is sucked out of the P-trap like that, dangerous sewer gases can leak into your kitchen.
You may be able to effect a temporary fix by snaking out the drain pipe downstream from the P-trap, but for safety's sake you really must get the drain vent cleared out.
The sink faucet has a leak and needs repair? The dishwasher drain line is connected usually with a barbed wye connector. It could just be the residual water left from when the dishwasher completed the drain cycle. Or water from the dishes in the dish rack?