Crankshaft position sensor reprogram

The short answer is, yes and no.

The long answer is that the ECM uses the crankshaft signal, along with cam position sensor in order to determine firing of the injection and ignition.

It also uses it to determine if the engine is misfiring by reading the amount of time between the signal pulses and comparing that to the data table it has been programmed with on how long those pulse separations should be.
Because of manufacturing changes and differences each sensor is slightly different in signal output and even in where it reads the signal depending on the position of the sensing part inside the actual head of the sensor.
The relearn procedure "resets" the sensor information so the ECM can learn the new sensors signal. If it is not done you can get mis-fire codes and poor engine performance or in newer engine a reduced power condition.

A dealer is not the only place that can do a re-learn, pretty much any good shop with a bi-directional scan tool can handle it as it is a very common item to be done, even for some diagnostic work.

On yours there is a thing called crank/cam correlation. Basically the ECM knows that the camshafts turn at half crank speed. It expects the cam sensor to send signal pulses in time with the crankshaft sensor so the proper cylinders get spark and fuel at the correct time. Your cam sensor is in the distributor and changing it does not require a relearn as it is a different style sensor than the crank sensor.

As the timing chain wears and the other valve train parts wear the cam signal can get "out of time" with the crank and you get a P1345 code. But the most common cause for that code on your engine (1996-1998 actually) is a worn distributor drive gear at the base of the distributor.

AutoZone will only read codes they will not do anything like resetting the calibration of the systems unless it's on a battery install where the new battery has to be initialized into the system.
To do the reset you need a proper scan tool with bi-directional controls so it would depend on what you have access to as to if it will work or not. However what are the actual codes you have in the system?

On a diesel with no warning lights I would suspect a fuel issue, like a partly clogged filter or a weak pump. Either would restrict fuel flow and give you those symptoms. However that vehicle isn't in out database so I cannot tell you how to test the pressure or the filters location, but it is possibly in the manual.