Lenovo T410s - 0190 critical low battery error

Hello,

I got old Lenovo T410s laptop which had very weak battery but I didn't care since I was always running on power adapter.
However suddenly it started to show:
0190 critical low battery error
during startup and turned off.
It contains additional battery in extension bay (i.e. CD bay).
I tried to remove this battery and it was very difficult.
It occurs this battery has swollen.
Removing additional battery and running with only primery battery didn't help. Still
0190 critical low battery error
error occurred and no go.
Removing primery battery with power adapter plugged in resulted in laptop not responding to power button at all.
One week later I've just checked and laptop stopped to responding to power button even with primery battery and power adapter plugged in.
Is the laptop dead?
Should the laptop run with batteries removed and power adapter plugged in?
Could the swollen battery in extension bay damage some important circuits in laptop?
Is there any chance that replacing primery battery will fix the issue?
Thanks.