Electrical issues are always a bugger to track down, especially on a car. Take this from a former Navy electrician who has owned many a beater car.

Most electrical issues involve grounding issues. Is the engine block properly grounded to the frame? If not then you will have issues as the block itself is part of the electrical circuit for your spark plugs.

Is it possible that your replacement alternator is bad? Without a good alternator, your car will only run until the battery is dead, which takes no more than a few minutes, and a bad alternator will destroy a new battery in short order.

Aside from that, the only other things I can think of involve using a multimeter and testing everything hither and yon, a task onerous enough that I would sell a kidney (preferably somebody else’s though) to pay a mechanic to do it for me. I had one car that had issues because the kid who owned it before me miswired the stereo in such a way to render the taillights utterly inoperative, so your gremlin may be somewhere other than the engine compartment.

I totally sympathize. I had a beater Buick that ate alternators for breakfast and it turned out to be faulty chip settings that nobody could control. If you had an older vehicle that could be garage-shopped, I’d say it’s worth finding a friend who likes to work on cars.

But you have computerized diagnostics involved. You’re better off saving your money and putting it down on a newer vehicle.
-From
the voice of experience and lots of lost cash

Check that your battery cable ends and battery posts are clean and oxide free and that your battery terminals are tightly attached to the cable.

I’m along with @jerv that it sounds like the alternator… or maybe alternator cables.
Somehow the power from the alternator isn’t making into the system.
Also the regulator could cause those symptoms – if it doesn’t properly detect that the alternator is putting out juice it would switch to the battery all the time which will wear out the battery pronto.

As a quick aside, the last time I blew a voltage regulator, my car did similar things near the end, as well as allowing the new battery to discharge so completely that it required replacement. And since most cars have the regulator built into the alternator….

My boss took a look at it and thinks the alternator is bad. But also that it could be a short in the wires going to or from the alternator. I’m gonna yank the alternator and have it tested, and then see what happens after that.

I’ll let you guys know what happens!