Hi @ashmlawson ,
There is always a reason for components overheating. Usually it is due to excessive current flow.
Replacing the 2 resistors may not fix the problem of why they both overheated in the first place.
Without a circuit diagram it might be difficult to find the real cause of the problem.
I suggest that you use an Ohmmeter to measure the resistance of the "coil" circuits in the blanket. Looking at the top picture there appears to be a 4 wire cable leading away from the board which I assume goes to the blanket.
To me this indicates at least 3 circuits in the blanket with a return path. (I may be wrong with all this of course). You may have to disconnect the wires off the board (mark the wires as to what wire goes where before you do so that you know how to reconnect them) and then measure the circuit. One may be significantly lower in resistance (even s/c perhaps) than the others and this is perhaps why there is more current flow than normal, overheating the 2 resistors on the board.
This is all supposition on my part and the fault may well be in the circuit board, but you have to start somewhere.
If there appears to be no problems with the blanket circuit then you'll have to trace the paths from the resistors to see what other components they initially connect to and check them for possible faults.
As I said earlier without a schematic not the easiest thing to do.