Update: I soldered a new plug onto the power adapter and babied that thing along all of this time with the help of tooth picks (and propping it up any way I could) in order to get the battery charged. I tried a couple of power adapters that were supposed to be "universal" that didn't work.
I even tried the correct HP power adapter that I borrowed from a friend that used to own a computer shop and it still had a problem.
Which meant there was a problem on the mother board with the DC. jack. (Which was what caused the plug to burn up on my adapter in the first place I discovered later.)
So I decided I was going to bite the bullet and buy a new laptop or tear that thing apart to see if I could fix it. (Or tear it up so badly nobody could fix it... what did I have to loose at that point?)
After finally getting the mother board out I discovered a cold solder joint that, according to my friend, may have been present when the thing was manufactured and just never became evident until recently.
Anyway, long story short, I managed to change the plugs to something I can buy at Rat Shack if this ever happens again. And everything works like it's supposed to as far as I can tell!
But, I learned some things during this operation:
1. My eyesight is piss poor.
2. Magnifying glasses are good things.
3. I have fat fingers.
4. Tweezers are good things.
5. Beer does NOT help the above!
When it's all said and done: I never want to see the inside of a laptop again!