Hi,
Yes it appears that the device that looks like a transistor has overheated and burnt out.
This can happen when the input voltage goes above the 10.8v it is meant to wrok with
normally. You might trace out the circuit to see how it may be fixed, or get a new one.
If you get a new one and you want it to work with a 12v battery that might be being
charged too, you have to drop the voltage down to 10.8 or something like that.
Remember though a 12v battery being charged can be as high as 14v or more.
Also, unless that number leads you to a specific device you would have to determine
if you need NPN, PNP, NMOSFET, or PMOSFET, or something like that.
You can also build your own charger that can work off of 12v.