Considering what you've already tried, you might need to be looking for something loading down the electrical system. My son had a frayed wire in a tail light assembly on a Cavaleir that caused a real similar issue.
Does the battery hold its charge? As rare as it is, I've also bought batteries and alternators that were bad out of the box.
If it was me, I'd first pull the alternator and battery and have them tested. Most alternators now have the main voltage regulator built in but if yous doesn't, that would be another area to check.
Good luck.
