|If the laptop only shuts down with it on battery power, but is ok with the mains lead plugged in, it must be the battery.|
As you have a new battery, it would seem the battery is alright. Unless the new battery is crap.
So, just because it says 60% left how do you know this is correct?
Trying different monitoring software can't do any harm. But then again its up to you.
I suppose it could be the battery connections.
A thank you would be nice, if I have helped.