@manaraja I have the exact same problem, right now, on my '91 LJ78 - single battery setup. I've confirmed with multimeter that the battery is not getting enough voltage while vehicle is running, which leads me to believe alternator is out. Voltage at battery drops to 11.8 when vehicle is on (with only radio and 3in1 gauge on), believe it should be around 14.2-14.7 if alternator is doing its job. 12.6v read on battery when vehicle is off is normal, so definitely needs to be above that to be getting charge from alternator. I've confirmed the battery (Diehard Red, only 1 yr old) is good. I'm hoping to have alternator refurbished and reinstalled. Right now I have the constant beep you describe, even when vehicle is off and key is out of ignition. Hoping alternator fix solves it, but hearing your experience makes me a bit nervous. I'll have to dig into wiring, fused links, and grounds a bit closer, too.
Did you get your issue resolved? Curious since we have exact same symptoms.