Help! My 2004, Lincoln Town Car seems to be having an electrical problem related to the battery! D:

I have a 2004, Lincoln town car. The first issue I came across was when I left
the light on accidentally, and it drained my battery to the point that I could
not start the car. It could be jump started. The mechanic determined that it
was a bad alternator combined with a bad battery. We self installed a new
alternator (it read 13.8 volts as opposed to the 12.8 the old one displayed)
and a new battery (which had 750 amps as opposed to the 700 amps the
old one had). The check engine light went off, as did the red battery light.
The issues went away for about 3 days and now the same symptoms are
back: The car won't start, the check engine light is on, and the red battery
light is on. Some of the smaller electrical functions inside the car work fine
for a short time but only if the car is parked. They don't work when the car is
being driven. There does not appear to be any corrosion on the battery
cables.