Okay, from the beginning: The Jeep went dead after a period of probably two weeks without being started. I jumped it and it worked for some time with being run consistently, but then died again. After this pattern being repeated for a few cycles, and then it not even being able to be jumped anymore, I had it towed to a local shop. They said it was the battery and alternator, which they replaced. It worked for a short time and the problem came back. I then took it to a place that specialized in electrical issues and they said the starter was shorted out and was draining the battery. It worked fine for some time after that until another extended period in which the Jeep was never started. When I went to jump it I noticed that the headlights came on immediately, without being switched on, and the Jeep wouldn't jump, which I assumed was because the lights were draining too much of the jump power (the battery is stone dead.) I also noticed that the lights stayed on until all power that had been charged into the battery from the jump process drained away.
I did a little research, which told me that the delay relay could be shorted out (the other alternative is the switch itself, which was much costlier.) So, I called the local Napa shop, which sold me a "delay relay" for about $11.
And that's where I'm at.
Thanks,
AB