Both headlight and blower switches as well as the blower resistors are the subject of recalls/extended warranty, depending on year of car. Check of VIN at dealer the best way to find out if your car is/was covered. The connectors seem to be the problem, causing overheating, as you have found out.
Blower circuit has +12 from blower relay contact fed to blower motor then to switch > resistors > ground. Hi-speed has no resistor as you are currently running. Early cars I believe have 3 speeds, mine have 4. If you turn ignition on you should measure volts to ground from all but one of the wires (probably black). With ignition off what you measured is correct.
My '97 service manual seems to have errors in this circuit, but as long as your new and old connectors match you should have no problems splicing in (assuming swich configuration is the same).
Please post again if you have any questions on above.
