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enough power to carry the electrical load and keep the battery charged up.
The poor acceleration and tendency to stall coupled with the 12 volt reading
is a sign of a discharged battery and low system voltage. The fact that it
takes 4 hours to get to this point indicates that the alternator is
generating some power.
Some potential causes, not in any order:
1. bad alternator/voltage regulator/brushes
2. corroded/dirty connections at the alternator/battery
3. bad grounds at the alternator/engine ground straps/battery ground
4. something in the A/C system drawing very high loads (witness improvement
with A/C off)
5. bad battery (even new batteries go bad)
6. faulty wiring in charging system
7. slipping belt (not too likely with the serpentine belt)