First of all, no offense, but your diagnosis-through-partswap kinda makes it sound like you're kinda new to electrical troubleshooting.
A properly-charging 12 volt automotive system should run between ~13.5-15.5 volts, and when the engine is off the battery should slowly (over hours) drop to 12.5-12.6 volts. So the battery being close to 14 volts with the engine running isn't a problem; it's ideal.
However, these trucks are notorious for having genuine overcharging (20v+) issues and no-charge issues.
To diagnose those, I first have to ask...
What year truck? Does it have a black rectangular cross-over tube over the engine that says "Cummins Turbo Diesel" or is it intercooled?
The earlier ones (89-91 with the cross-over tube) don't have an "ECM" that controls charging voltage, they have a standalone solid-state "voltage regulator" that mounts on the firewall, does the same thing and is much cheaper. You can actually retrofit the earlier-style voltage regulator if the ECM stops working on a 91-93 intercooled truck.
Voltage regulators and ECM's like to go bad on these trucks. A lot of the time that manifests as an intermittent overcharging problem.
Our alternators are speshule; they get an ignition-on 12v+ source from a blue wire, then either the voltage regulator or the ECM gives it a variable 5-7 volt signal through a green wire. The volt drop across the field coil in the alternator is functionally equal to the blue wire voltage minus the green wire voltage; usually this is between 7 or 5 volts. If the alternator stops getting a good 12 volt signal from the blue wire, it won't do anything. If it stops getting a 5-7 volt signal from the green wire it will full field the alternator and, depending on the engine RPM, it can kick out voltages over 30 volts and cause all sorts of havoc. So overcharging is usually caused by the ECM or voltage regulator not putting out enough voltage to the alternator, or becoming disconnected from the alternator.
