Ashley; Your computer depends upon a constant 12V for it to function. If the battery/charging system drops below that point, it will go into what's called 'Limp home mode'. The engine will 'run' but not very well. The computer relies on information from various sensors to control fuel delivery. The Coolant temp sensor is a very important one. If the coolant level gets low, it will 'tell' the computer that the engine is cold, whether it really is or not. If the engine got hot when the radiator 'blew', it is possible that the thermostat was damaged and/or the sensor. The cooling fans are there for two reasons. The main fan comes on when the engine reaches a specified temp, usually around 230F. The auxilary fan will come on when you have the heater/AC control set to defrost or A/C.
Back to the radiator; The coolant temp sensor must be in physical contact with the coolant/water. Hopefully, this isn't the case, but if you got the engine hot enough, you may have damaged the head gasket which will allow exhaust gasses into the cooling system, resulting in air pockets which will give the CTS erroneous readings. And, overheating issues. Again, hopefully, this isn't the case.
Loren
SW WA