Replacing the socket itself should be fairly easy. You should be able to just pry the cover off and then remove the wires with their pins attached. Then you place the wires and pins into the new socket in their proper locations and snap the cover on. The part numbers should be on the socket and the plug. Most likely their will be a separate part number for the socket and the plug. The total cost for both will probably be less than $10.00.

I do have one concern, however. When the heat built up in the wiring enough to char your connector it may have also made the insulation on the wiring brittle. Check to make sure that none of the insulation toward the end of the wiring is hard or, worse yet, broken off. You may need to trim it back and put new ends on it if that is the case.