Yep, 11.94 Vs 11.97 can be accounted for by 1A flowing through a 0.03 ohm resistance. A length of PCB track will have that.
You're right, the 7.79 volts is interesting because that point should not get below about 11.3V. This suggests to me that this transistor is toast.
However, before we come to that final conclusion, place your meter on it's diode check range (or resistance range if you don't have a diode test range), and with the power off, and the red probe on the point which measured 7.79V, and the black probe on the one which measured 11.94, tell me what you see (and the range it's on)
Then reverse the probes and try again.
Just to give these point their proper names... 11.94 is the emitter (E), 0.1V is the collector (C), and 7.79 is the base (B).
I have asked you to give me readings both ways across B and E.
It would also be useful to get the readings (again, both ways) across B and C, and C and E
The best way to give me the results is in a table:
Diode test/resistance range (specify)
Black B, Red E reading
Red E, Black B reading
Black B, Red C reading
Red C, Black B reading
Black C, Red E reading
Red E, Black C reading