67 Barracuda Heavy Electric Load
I guarantee to you that if the battery is registering 12.22 volts when the motor is running, that's because the alternator is dead.
Something has expired between the alternator, the voltage regulator, maybe a fuse somewhere, or the wiring got nibbled by a porcupine in the dead of night!
I think you meant that "THE CHARGING SYSTEM" is dead.
NOTE THAT the initial statement is misleading. Either the charging system is dead or the battery is way down OR defective and will not "come up.
But it does not mean (as explained) that "the alternator" per se is the culprit
I would temporarily swap in a known good and charged battery. KNOWN!!
I would figure a way to load the system to max amperage, heater, lights, etc, AND FANS
If the voltage will not regulate to nominal 14V, then "full field" the alternator, that is, bypass the regulator
Do this first by removing the green field wire from the alternator, and connect a jumper from that now exposed alternator field terminal to ground. Watch the voltage and do not allow it to increase above about 16V
If it will not, remove and inspect alternator. READ the service manual in the electrical section concerning field current draw test. Remove and inspect brushes for wear, spring and free movement, debri, grease and oil.
The main problems with a low output alternator are:
Poor brush contact, or field winding problems
Bad diodes, either 1 or more
Stator winding problems, which can be a partial open in one winding, partial shorts, etc etc. Sometimes you can SEE these physically. Burned toasted wires, loosened up from vibrations and rubbing, etc etc.