There is a very simple answer to your question. Your dishwasher is draining into the side drain adapter on your garbage disposer. this was a common practice for many years. All the water draining from your dishwasher is being pumped into the upper basin side of the disposer. Solids from the dishwasher are blocking the drain in the disposer, so when you turn on the disposer, it grinds the waste and drains the water. This is actually the way it is suppose to work. Current codes require a separate trap for the dishwasher and not plumbed into the disposer. If you look at the side of your disposer, you will see the hose from the washer attached there. Only fix is to install a new separate trap and drain for the washer. Good luck.
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.
Best Answer
Sounds like a plugged drain which might be in the outlet of the disposer, but most probably is below where the drains join, maybe in the U-trap. Fill both sinks 1/4 full of water and use a plunger first on the non-disposer sink and see if this clears the blockage. If it doesn't, then use the plunger on the disposer side. Push down easy and pull up vigorously on the plunger to pull the plug out and apart, then see if this clears the blockage. Put a stopper in the other drain when you are plunging on one (prevents the suction action of the plunger from just pulling in water through the other drain). The action you want it to pull on the blockage from below.
You may have pieces of wood lodged in the drain. If solid material comes up into the sink, fish it out and put it in the trash.