I use to have a connector just like that.
A new connector will/should come with new crimp terminals. Just make sure you crimp them well. There's enough wire to cut the melted stuff off, strip and recrimp so I don't think heat shrink is necessary. My thinking is the switch current capacity is really to low for the application. Over time the contacts are eroded from the sudden surge of current through them when you turn on the fan. Over time this gets worse as the switch loses its current carrying capacity, the heat is conducted through the terminals where it melts the connector.I've considered having the switch control some relays to do this. Maybe someone has already done it. Another idea would be a PWM controller for fan speed instead. Modern cars do this so there might be something we could use.