OK the P0622 code is, as stated, The Generator Field Control Circuit.
Now it would seem that your original problem was that the battery was not charging at all hence the lights going dim and then not starting.
My big concern is that your replacement alternator may be faulty or may be the wrong PN for your 2003 Jeep. The 2003 Jeep Alternator has a field coil built inside it...this is the small 2 pin connector on the alternator. One pin, pin #2, goes to +12 volts on the PCM, the White/Dark Blue wire....check that +12 volts is there! The pin #1 Dark Green wire also goes to the PCM.
The PCM reads the voltage of the battery and adjusts the current output by putting negative pulses of varying length on pin#1. This is called Pulse Width Modulation (PWM). The longer this negative pulse is the greater the voltage put out by the alternator.
So if pin #1 Dark Green wire is shorted to engine/chassis it means the alternator will put out maximum power which is what you are getting. So you need to disconnect the battery and pull plug # C2 off the PCM and pull the small plug off the alternator.
So this Dark Green wire should be hanging in mid air...test with a Muti-Meter that it is not in fact shorting to chassis which would cause the generator output to go to maximum.
Different year KJs use different methods to adjust the generator output voltage which is why you must have the correct Part Number! I will check into this but maybe you should replace the bearings in your original alternator and use it.
You can download the 2003 Jeep KJ Service Manual here.
Section 8F describes how the alternator works and Section 8W has the wiring diagrams...your main interest is page 8w-20-2.
These wiring problems can be difficult to find! You may have blown some fuses with the higher voltage!
www.colorado4wheel.com/manuals/Jeep/KJ/
OK the 2006 Jeep KJ 3.7 gasser wiring diagram shows the same type of charging theory to the 2003 wiring diagram but pins 1 and 2 are the other way around. This is not unusual for the wiring diagrams to have errors. So maybe pin#2 the White/Dark blue wire is the one that gets switched to ground and this wire is the one that needs to be checked for a short to ground!