Hi, proscrastin8, and welcome to Apple Discussions. If you've tried three different AC adapters, at least one of which is known to be good, and none of them charges your battery, you have an internal hardware problem in your Powerbook that needs repair.
Your Powerbook won't start up because whatever is wrong prevents it from being powered by the AC adapter, just as it prevents the battery from being charged. When the Powerbook seems to be trying to start up, then abruptly goes to sleep, it's because your nearly-dead main battery has just enough juice left (probably after some rest time) to begin the startup process, but not nearly enough to see it through.
You can hope the problem is a bad DC-in card or a bad connection from that card to the logic board, if you have one of the Tibook models in which that card is separate from the logic board. The card is easy to replace in those models, and relatively cheap. If that isn't it, you're probably facing a logic board replacement, which may or may not be worth doing.
Which Tibook model do you have?