Won't start again; WTF?

1996 Ram 1500 SLT Laramie 4x4 5.9 I use it very infrequently. I let it sit for about a month, and then it wouldn't start. No lights, no buzzers, nothing. I jumped it from
another truck and got it to start, but every time I let my foot off the gas, it died. So I'm thinking: bad charging problem + a fuel problem. OK.
So last Wednesday (today is the following Thursday) I charged the battery and replaced the fuel pump relay (started with the cheapest fix first), and whaddayaknow, everything worked fine again. I drove it around for that evening, and everything was cool. I took it in for state inspection Thursday night and picked it up on Saturday morning. Drove it Saturday, and drove it Sunday.
Today I tried to start it up, and paragraph one happened all over again. My guess is I was running on battery power all those days (but that seems pretty long) because of a bad alternator. But it does not make any sense to me as to why I would have the fuel delivery problem again. I suppose it's possible that I got a bad relay, or that my fuel pump decided to bail on me at a very ironic moment, but the symptoms that I fixed in paragraph two returning exactly as they were in paragraph one, I find strange.
Does anyone have any ideas, or seen this before?

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