It is important that the cord be rated for a high amperage......20+ amps
PC Power cords are typically 3 conductor18AWG ..... I certainly would NOT notch out a computer power cord to make it fit. It may work for a while but it will melt.
The power cord you need should be at least 14 AWG Better would be 12AWG
http://www.stayonline.com/reference-circuit-ampacity.aspxAny loose connection in the circuit will result in heat at that connection. There may be something loose or a inferior connection between the actual plug on the unit and the heating element. In which case , I would suggest that the smoker is defective.
If the cable is too small ( i.e.internal cord wire is 20AWG or 18AWG ) the whole cable would get warm and could melt or catch fire.... BEFORE your circuit breaker trips
The cable should be similar to those used in Kettles etc.. (R151M plug) Get good quality. The female end should NOT be a molded end. It should be able to be opened up to check the wire connections...
The best ( safest ) route even with a high amp rated cable would be to also use a GFI outlet
The fact that your previous cord melted to the point that metal parts became exposed shows how close you came to having a fire on your hands.
Dont second guess this stuff .... specs matter.