Short story: When main power battery is installed & AC charger is also connected, it doesn't work. If battery is removed, it works fine with charger only. Anyone has any idea why & how battery is causing this? Is there any way to fix this battery?

More info:
-Battery & charger are several years old.
-OS: Windows 2000 Prof., SP4, BIOS: 1.22
-I always used it with battery on the laptop & charger connected, because with battery only, it lasted only for few minutes (not a good battery). But I kept the battery on the unit, for just in case there was a sudden power outage, I have few minutes to save my stuff.
-With both battery & charger connected, everything used to work till recently, when suddenly the monitor went off (middle of using it), while I still could hear the HD spinning but more quietly (not like when it’s actively running & processing). I tried everything but monitor didn’t come up. I had to push & hold the power switch for few seconds to shut it down. Pushed the switch again: while it was coming up, the same thing happened (& every time after).
-I removed the power battery, disconnected the charger, disconnected the CMOS battery, and removed HD, RAM, etc (all together) for several hours & put them back: but it didn’t help.
-When everything is cool, it comes up for few minutes before it hangs. More I try (which makes it warmer), up time becomes shorter & shorter, to a point that when I turn it on, after some seconds (while its booting), it hangs.
-Tried the same HD, battery, & charger on another T20: same problem.
-Tried with another HD: same problem.
-When I turn it on, initially, while in the 1st screen (with ThinkPad logo, & F1 & F12 options at the bottom of the screen), fan is on but right after it passes that screen (which means it starts the start-up process), fan goes off. I thought maybe it gets overheated. I checked the fan, frame, thermal pad, etc: they seem fine.
-Tried with putting an ice-pad under the laptop: it lasted few more minutes but same thing happened again.
-Remove the power battery & just use the charger: no problem – it works fine.
-If I put the battery on (with charger): same issue happens again.

I understand that the battery is old. I thought, as long as the AC charger is connected, the battery doesn’t play any significant role. I appreciate if anyone knows why & how battery can cause this? And, is there any way to fix this battery?
Thanks.