toshiba chromebook 2 turns off when unplugged, new battery

Solution
Then I would be looking at the battery. Is there something wrong with it. If it is aftermarket then that is a likely issue, as with those it is hit or miss. You may want to consider returning it and getting an OEM one.
Even with a new battery, if it isn't charging then that won't work. And, if it is an aftermarket battery it may not work either.

Try turning off the laptop and see if it will charge then. If it charges when off, but not when turned off, then the charger is the problem. It isn't putting out enough power to do both and would need replacing.

However, if it won't charge no matter if off or on, but the laptop runs when plugged in, that battery (new or not) is the problem.
 

Battery is full 100% turns off when unplugged