My 2001 Dodge Ram 1500 will start but dies while idling, what's wrong?

I have the 5.9 L 1500, and my battery had been drained completely a week ago when I left my lights on. Since then, when I go to start my truck, it turns over like the battery is low, but has always started. Recently, however, it dies right after I start it. I have held the gas and brake down, and driven it around the block, and then when I get back to my place it will be idling just fine. Also, after driving it around for a bit, I can shut it off and fire it right back up with no problems (it turns over like the battery is fully charged, and idles just fine.) What the heck could be wrong with it? I cleaned the battery leads and wires, but I don't know what is going on. There was a lot of corrosion on the positive terminal, but it's all clean now. could a bad battery cause this? I had no issues before it was drained completely by my lights on overnight. That is my initial thought, because it will fire up just fine right after I drive, but it will struggle a bit after an overnight parking.